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Subfields Splitting Fields Adjoining Elements Field Extensions and Splitting Fields Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields Subfields


  1. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field of characteristic zero and let p ∈ F [ x ] be a polynomial of positive degree. The equation p ( x ) = 0 is solvable by radicals iff all its solutions can be calculated from its coefficients in a finite number of steps using field operations (addition, multiplication, additive and multiplicative inversion) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  2. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field of characteristic zero and let p ∈ F [ x ] be a polynomial of positive degree. The equation p ( x ) = 0 is solvable by radicals iff all its solutions can be calculated from its coefficients in a finite number of steps using field operations (addition, multiplication, additive and multiplicative inversion) and root extractions. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  3. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field of characteristic zero and let p ∈ F [ x ] be a polynomial of positive degree. The equation p ( x ) = 0 is solvable by radicals iff all its solutions can be calculated from its coefficients in a finite number of steps using field operations (addition, multiplication, additive and multiplicative inversion) and root extractions. The root extractions are allowed to yield elements that are not in F , but in an extension field E of F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  4. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field of characteristic zero and let p ∈ F [ x ] be a polynomial of positive degree. The equation p ( x ) = 0 is solvable by radicals iff all its solutions can be calculated from its coefficients in a finite number of steps using field operations (addition, multiplication, additive and multiplicative inversion) and root extractions. The root extractions are allowed to yield elements that are not in F , but in an extension field E of F . Note that solvability by radicals does not mean there is a general formula. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  5. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field of characteristic zero and let p ∈ F [ x ] be a polynomial of positive degree. The equation p ( x ) = 0 is solvable by radicals iff all its solutions can be calculated from its coefficients in a finite number of steps using field operations (addition, multiplication, additive and multiplicative inversion) and root extractions. The root extractions are allowed to yield elements that are not in F , but in an extension field E of F . Note that solvability by radicals does not mean there is a general formula. It means that there is some way to express the zeros of the polynomial under investigation . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  6. Subfields Splitting Fields Adjoining Elements Proposition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  7. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family of subfields of E . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  8. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  9. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  10. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  11. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  12. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J operations in E logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  13. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  14. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  15. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover, j ∈ J associativity logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  16. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover, j ∈ J associativity, commutativity logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  17. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover, j ∈ J associativity, commutativity and distributivity of multiplication � over addition hold in F j , too. j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  18. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover, j ∈ J associativity, commutativity and distributivity of multiplication � over addition hold in F j , too. 0 and 1 are identity elements j ∈ J for addition and multiplication in E . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  19. Subfields Splitting Fields Adjoining Elements Proposition. Let ( E , + , · ) be a field and let { F j } j ∈ J be a family � of subfields of E . Then F j is a subfield of E . j ∈ J � Proof. By assumption, 0 , 1 ∈ F j . Because the F j are j ∈ J subfields, sums and products of elements of F j are in F j , too. � The binary operations on F j are the restrictions of the binary j ∈ J � operations in E and they map into F j . Moreover, j ∈ J associativity, commutativity and distributivity of multiplication � over addition hold in F j , too. 0 and 1 are identity elements j ∈ J for addition and multiplication in E . Hence they are the identity � F j . elements for addition and multiplication in j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  20. Subfields Splitting Fields Adjoining Elements Proof (concl.). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  21. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  22. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  23. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  24. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then y logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  25. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then y = y + 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  26. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  27. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  28. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  29. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) = − x . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  30. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) = − x . � Because j ∈ J was arbitrary, we conclude that − x ∈ F j . j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  31. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) = − x . � Because j ∈ J was arbitrary, we conclude that − x ∈ F j . Thus j ∈ J � F j contains additive inverses. j ∈ J logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  32. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) = − x . � Because j ∈ J was arbitrary, we conclude that − x ∈ F j . Thus j ∈ J � F j contains additive inverses. Multiplicative inverses are j ∈ J handled similarly. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  33. Subfields Splitting Fields Adjoining Elements � Proof (concl.). For inverses, let x ∈ F j . Then x has a unique j ∈ J additive inverse − x in E . Let j ∈ J and let y be the additive inverse of x in F j . Then � � y = y + 0 = y + x +( − x ) = ( y + x )+( − x ) = 0 +( − x ) = − x . � Because j ∈ J was arbitrary, we conclude that − x ∈ F j . Thus j ∈ J � F j contains additive inverses. Multiplicative inverses are j ∈ J handled similarly. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  34. Subfields Splitting Fields Adjoining Elements Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  35. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  36. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . Then f splits in the extension field E ⊇ F iff p can be factored into linear factors with coefficients in E [ x ] . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  37. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . Then f splits in the extension field E ⊇ F iff p can be factored into linear factors with coefficients in E [ x ] . Now let F be a field, let p ∈ F [ x ] and let E be an extension field in which p splits. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  38. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . Then f splits in the extension field E ⊇ F iff p can be factored into linear factors with coefficients in E [ x ] . Now let F be a field, let p ∈ F [ x ] and let E be an extension field in which p splits. Then the field � S : = { D : D is an extension field of F , D ⊆ E , p splits in D } logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  39. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . Then f splits in the extension field E ⊇ F iff p can be factored into linear factors with coefficients in E [ x ] . Now let F be a field, let p ∈ F [ x ] and let E be an extension field in which p splits. Then the field � S : = { D : D is an extension field of F , D ⊆ E , p splits in D } is called the splitting field for p over F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  40. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let p ∈ F [ x ] be a polynomial over F and let E be an extension of F . Then f splits in the extension field E ⊇ F iff p can be factored into linear factors with coefficients in E [ x ] . Now let F be a field, let p ∈ F [ x ] and let E be an extension field in which p splits. Then the field � S : = { D : D is an extension field of F , D ⊆ E , p splits in D } is called the splitting field for p over F . (We will address the fact that we say “the” in the next presentation.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  41. Subfields Splitting Fields Adjoining Elements Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  42. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  43. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  44. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  45. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  46. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  47. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  48. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  49. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  50. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  51. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using sums logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  52. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using sums, products logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  53. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using sums, products, additive inversions logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  54. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using sums, products, additive inversions and multiplicative inversions logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  55. Subfields Splitting Fields Adjoining Elements Definition. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . We define F ( θ 1 ,..., θ n ) to be the intersection of all subfields of E that contain F and θ 1 ,..., θ n . Then F ( θ 1 ,..., θ n ) is called the field F with the elements θ 1 ,..., θ n adjoined . Example. C = R ( i ) . Theorem. Let ( F , + , · ) be a field, let E be an extension of F and let θ 1 ,..., θ n ∈ E \ F . Then the elements of F ( θ 1 ,..., θ n ) are rational combinations of the θ j , where a rational combination is formed from elements of F and the θ 1 ,..., θ n using sums, products, additive inversions and multiplicative inversions (except divisions by zero). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  56. Subfields Splitting Fields Adjoining Elements Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  57. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  58. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  59. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  60. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  61. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  62. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  63. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  64. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  65. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  66. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. By induction hypothesis, for j = 1 , 2 we have r j = p j , where p j and q j q j are polynomial combinations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  67. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. By induction hypothesis, for j = 1 , 2 we have r j = p j , where p j and q j q j are polynomial combinations. Now r logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  68. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. By induction hypothesis, for j = 1 , 2 we have r j = p j , where p j and q j q j are polynomial combinations. Now r = r 1 + r 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  69. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. By induction hypothesis, for j = 1 , 2 we have r j = p j , where p j and q j q j are polynomial combinations. Now r = r 1 + r 2 = p 1 + p 2 q 1 q 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  70. Subfields Splitting Fields Adjoining Elements Proof. A polynomial combination is formed from the elements of F and θ 1 ,..., θ n using sums, products and additive inversions. We first prove by induction on the total number k of operations (sums, products, additive and multiplicative inversions) needed to form a rational combination r that r = p q , where p and q are polynomial combinations. k = 0: Trivial: r ∈ F ∪{ θ 1 ,..., θ n } and r = r 1. Induction step, k > 0 : Let r be a rational combination. First case: r = r 1 + r 2 , where r 1 and r 2 are rational combinations. Both r 1 and r 2 were formed using fewer than k operations. By induction hypothesis, for j = 1 , 2 we have r j = p j , where p j and q j q j are polynomial combinations. Now = p 1 q 2 + p 2 q 1 r = r 1 + r 2 = p 1 + p 2 . q 1 q 2 q 1 q 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  71. Subfields Splitting Fields Adjoining Elements Proof (concl.). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  72. Subfields Splitting Fields Adjoining Elements Proof (concl.). The arguments for r = r 1 · r 2 , r = − r 1 and r = ( r 1 ) − 1 (for r 1 � = 0) are similar. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  73. Subfields Splitting Fields Adjoining Elements Proof (concl.). The arguments for r = r 1 · r 2 , r = − r 1 and r = ( r 1 ) − 1 (for r 1 � = 0) are similar. It is now easy to verify that the rational combinations are a field logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  74. Subfields Splitting Fields Adjoining Elements Proof (concl.). The arguments for r = r 1 · r 2 , r = − r 1 and r = ( r 1 ) − 1 (for r 1 � = 0) are similar. It is now easy to verify that the rational combinations are a field and that every subfield of E that contains F ∪{ θ 1 ,..., θ n } contains all rational combinations of the θ j . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  75. Subfields Splitting Fields Adjoining Elements Proof (concl.). The arguments for r = r 1 · r 2 , r = − r 1 and r = ( r 1 ) − 1 (for r 1 � = 0) are similar. It is now easy to verify that the rational combinations are a field and that every subfield of E that contains F ∪{ θ 1 ,..., θ n } contains all rational combinations of the θ j . (Good exercise.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  76. Subfields Splitting Fields Adjoining Elements Proof (concl.). The arguments for r = r 1 · r 2 , r = − r 1 and r = ( r 1 ) − 1 (for r 1 � = 0) are similar. It is now easy to verify that the rational combinations are a field and that every subfield of E that contains F ∪{ θ 1 ,..., θ n } contains all rational combinations of the θ j . (Good exercise.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

  77. Subfields Splitting Fields Adjoining Elements Proposition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Field Extensions and Splitting Fields

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