Preliminaries The Witt Ring Denote by H the principal ideal of � W ( K ) generated by [ H ]. The Witt Ring of K is defined as W ( K ) := � W ( K ) / H . If ϕ is a quadratic form over K , then { ϕ } will denote its equivalence class in W ( K ). Note The elements of W ( K ) classify anisotropic quadratic forms over K . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 10 / 32
Annihilating Polynomials Annihilating Polynomials Consider the canonical inclusion → � ι : Z − W ( K ) defined for m ∈ N by m �− → [ m × � 1 � ] . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 11 / 32
Annihilating Polynomials Annihilating Polynomials Consider the canonical inclusion → � ι : Z − W ( K ) defined for m ∈ N by m �− → [ m × � 1 � ] . Usually we will simply write m for its image via ι in � W ( K ). Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 11 / 32
Annihilating Polynomials Annihilating Polynomials Consider the canonical inclusion → � ι : Z − W ( K ) defined for m ∈ N by m �− → [ m × � 1 � ] . Usually we will simply write m for its image via ι in � W ( K ). Definition A polynomial P = z n X n + · · · + z 0 ∈ Z [ X ] is called annihilating polynomial of a quadratic form ϕ over K , if P ([ ϕ ]) := z n [ ϕ ] n + · · · + z 1 [ ϕ ] + z n = 0 ∈ � W ( K ) . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 11 / 32
Annihilating Polynomials Annihilating Polynomials Consider the canonical inclusion → � ι : Z − W ( K ) defined for m ∈ N by m �− → [ m × � 1 � ] . Usually we will simply write m for its image via ι in � W ( K ). Definition A polynomial P = z n X n + · · · + z 0 ∈ Z [ X ] is called annihilating polynomial of a quadratic form ϕ over K , if P ([ ϕ ]) := z n [ ϕ ] n + · · · + z 1 [ ϕ ] + z n = 0 ∈ � W ( K ) . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 11 / 32
Annihilating Polynomials The Lewis Polynomials In 1987 David Lewis introduced the following annihilating polynomials: For n ∈ N 0 he defined P n := ( X − n )( X − n + 2) · · · ( X + n − 2)( X + n ) ∈ Z [ X ] . He furthermore proved that P n annihilates all quadratic forms of dimension n over an arbitrary field K . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 12 / 32
Annihilating Polynomials The Lewis Polynomials In 1987 David Lewis introduced the following annihilating polynomials: For n ∈ N 0 he defined P n := ( X − n )( X − n + 2) · · · ( X + n − 2)( X + n ) ∈ Z [ X ] . He furthermore proved that P n annihilates all quadratic forms of dimension n over an arbitrary field K . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 12 / 32
Annihilating Polynomials The Lewis Polynomials In 1987 David Lewis introduced the following annihilating polynomials: For n ∈ N 0 he defined P n := ( X − n )( X − n + 2) · · · ( X + n − 2)( X + n ) ∈ Z [ X ] . He furthermore proved that P n annihilates all quadratic forms of dimension n over an arbitrary field K . These polynomials constitute the base for all our following observations. Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 12 / 32
Annihilating Polynomials The Embracing Polynomial Proposition For every quadratic form ϕ over K there exists a unique polynomial Q ϕ ∈ Z [ X ] which divides all annihilating polynomials of ϕ and has maximal degree among all polynomials with this properties. Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 13 / 32
Annihilating Polynomials The Embracing Polynomial Proposition For every quadratic form ϕ over K there exists a unique polynomial Q ϕ ∈ Z [ X ] which divides all annihilating polynomials of ϕ and has maximal degree among all polynomials with this properties. This polynomial Q ϕ is monic. Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 13 / 32
Annihilating Polynomials The Embracing Polynomial Proposition For every quadratic form ϕ over K there exists a unique polynomial Q ϕ ∈ Z [ X ] which divides all annihilating polynomials of ϕ and has maximal degree among all polynomials with this properties. This polynomial Q ϕ is monic. Furthermore there exists some m ∈ N such that mQ ϕ is an annihilating polynomial of ϕ . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 13 / 32
Annihilating Polynomials The Embracing Polynomial Proposition For every quadratic form ϕ over K there exists a unique polynomial Q ϕ ∈ Z [ X ] which divides all annihilating polynomials of ϕ and has maximal degree among all polynomials with this properties. This polynomial Q ϕ is monic. Furthermore there exists some m ∈ N such that mQ ϕ is an annihilating polynomial of ϕ . Idea of Proof. Use the greatest common divisor and B´ ezout’s identity. Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 13 / 32
Annihilating Polynomials Annihilating Ideals Definition The Q ϕ from the proposition is called embracing polynomial of ϕ . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 14 / 32
Annihilating Polynomials Annihilating Ideals Definition The Q ϕ from the proposition is called embracing polynomial of ϕ . The ideal Ann ϕ ⊂ Z [ X ] of all annihilating polynomials of ϕ is called annihilating ideal of ϕ . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 14 / 32
Annihilating Polynomials Annihilating Ideals Definition The Q ϕ from the proposition is called embracing polynomial of ϕ . The ideal Ann ϕ ⊂ Z [ X ] of all annihilating polynomials of ϕ is called annihilating ideal of ϕ . Q ϕ is called embracing polynomial since Ann ϕ ⊂ ( Q ϕ ). Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 14 / 32
Annihilating Polynomials A Common Factor Proposition For every n -dimensional quadratic form ϕ over K , the factor X − n ∈ Z [ X ] divides Q ϕ . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 15 / 32
Annihilating Polynomials A Common Factor Proposition For every n -dimensional quadratic form ϕ over K , the factor X − n ∈ Z [ X ] divides Q ϕ . Sketch of Proof. Recall that Q ϕ is a product of linear factors. The claim follows since dim : � W ( K ) → Z is a ring homomorphism and Z is an integral domain. Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 15 / 32
Annihilating Polynomials Examples Let ϕ be a quadratic form over K . K Pythagorean. Then � W ( K ) is torsion free. = ⇒ We have Q ϕ ([ ϕ ]) = 0 and therefore Ann ϕ = ( Q ϕ ). Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 16 / 32
Annihilating Polynomials Examples Let ϕ be a quadratic form over K . K Pythagorean. Then � W ( K ) is torsion free. = ⇒ We have Q ϕ ([ ϕ ]) = 0 and therefore Ann ϕ = ( Q ϕ ). Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 16 / 32
Annihilating Polynomials Examples Let ϕ be a quadratic form over K . K Pythagorean. Then � W ( K ) is torsion free. = ⇒ We have Q ϕ ([ ϕ ]) = 0 and therefore Ann ϕ = ( Q ϕ ). K not formally real. Then an element of � W ( K ) is torsion if and only if its dimension is 0. = ⇒ Q ϕ = X − n ∈ Z [ X ] Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 16 / 32
Annihilating Polynomials Examples Let ϕ be a quadratic form over K . K Pythagorean. Then � W ( K ) is torsion free. = ⇒ We have Q ϕ ([ ϕ ]) = 0 and therefore Ann ϕ = ( Q ϕ ). K not formally real. Then an element of � W ( K ) is torsion if and only if its dimension is 0. = ⇒ Q ϕ = X − n ∈ Z [ X ] Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 16 / 32
Formally Real Fields Formally Real Fields In this section K is a formally real field. X K is the space of orderings of K . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 17 / 32
Formally Real Fields Formally Real Fields In this section K is a formally real field. X K is the space of orderings of K . For each A ∈ X K there exists a signature homomorphism W ( K ) − → Z defined by � 1 if a > A 0 , {� a �} �− → − 1 otherwise . This homomorphism will be denoted by sign A . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 17 / 32
Formally Real Fields The Signature Polynomial Let ϕ be a quadratic form of dimension n over K . Set S sign := { sign A ( { ϕ } ) | A ∈ X K } ∪ { n } ⊂ Z . ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 18 / 32
Formally Real Fields The Signature Polynomial Let ϕ be a quadratic form of dimension n over K . Set S sign := { sign A ( { ϕ } ) | A ∈ X K } ∪ { n } ⊂ Z . ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 18 / 32
Formally Real Fields The Signature Polynomial Let ϕ be a quadratic form of dimension n over K . Set S sign := { sign A ( { ϕ } ) | A ∈ X K } ∪ { n } ⊂ Z . ϕ We have − n ≤ sign A ( { ϕ } ) ≤ n for all A ∈ X K . ⇒ S sign = is finite. ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 18 / 32
Formally Real Fields The Signature Polynomial Let ϕ be a quadratic form of dimension n over K . Set S sign := { sign A ( { ϕ } ) | A ∈ X K } ∪ { n } ⊂ Z . ϕ We have − n ≤ sign A ( { ϕ } ) ≤ n for all A ∈ X K . ⇒ S sign = is finite. ϕ = ⇒ We can define � P sign := ( X − s ) ∈ Z [ X ] . ϕ s ∈ S sign ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 18 / 32
Formally Real Fields The Signature Polynomial Let ϕ be a quadratic form of dimension n over K . Set S sign := { sign A ( { ϕ } ) | A ∈ X K } ∪ { n } ⊂ Z . ϕ We have − n ≤ sign A ( { ϕ } ) ≤ n for all A ∈ X K . ⇒ S sign = is finite. ϕ = ⇒ We can define � P sign := ( X − s ) ∈ Z [ X ] . ϕ s ∈ S sign ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 18 / 32
Formally Real Fields The Embracing Polynomial over Formally Real Fields Theorem If ϕ is an n -dimensional quadratic form over a formally real field K , then Q ϕ = P sign . ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 19 / 32
Formally Real Fields The Embracing Polynomial over Formally Real Fields Theorem If ϕ is an n -dimensional quadratic form over a formally real field K , then Q ϕ = P sign . ϕ Proof. We have dim( P sign ([ ϕ ])) = 0 and sign A ( P sign ([ ϕ ])) = 0 for all a ∈ X K . ϕ ϕ ⇒ P sign = ([ ϕ ]) is torsion. ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 19 / 32
Formally Real Fields The Embracing Polynomial over Formally Real Fields Theorem If ϕ is an n -dimensional quadratic form over a formally real field K , then Q ϕ = P sign . ϕ Proof. We have dim( P sign ([ ϕ ])) = 0 and sign A ( P sign ([ ϕ ])) = 0 for all a ∈ X K . ϕ ϕ ⇒ P sign = ([ ϕ ]) is torsion. ϕ Since Q ϕ ([ ϕ ]) is torsion and Q ϕ is a product of linear factors ⇒ P sign = divides Q ϕ . ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 19 / 32
Formally Real Fields The Embracing Polynomial over Formally Real Fields Theorem If ϕ is an n -dimensional quadratic form over a formally real field K , then Q ϕ = P sign . ϕ Proof. We have dim( P sign ([ ϕ ])) = 0 and sign A ( P sign ([ ϕ ])) = 0 for all a ∈ X K . ϕ ϕ ⇒ P sign = ([ ϕ ]) is torsion. ϕ Since Q ϕ ([ ϕ ]) is torsion and Q ϕ is a product of linear factors ⇒ P sign = divides Q ϕ . ϕ ⇒ P sign = = Q ϕ ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 19 / 32
Formally Real Fields Example Let ϕ be a quadratic form over R with n = dim( ϕ ). Set s := sign( { ϕ } ) (there exists only one ordering of R ). Since R is Pythagorean, we have already seen that Ann ϕ = ( Q ϕ ). Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 20 / 32
Formally Real Fields Example Let ϕ be a quadratic form over R with n = dim( ϕ ). Set s := sign( { ϕ } ) (there exists only one ordering of R ). Since R is Pythagorean, we have already seen that Ann ϕ = ( Q ϕ ). Hence by the theorem � ( X − n ) if s = n , Ann ϕ = (( X − s )( X − n )) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 20 / 32
Formally Real Fields Example Let ϕ be a quadratic form over R with n = dim( ϕ ). Set s := sign( { ϕ } ) (there exists only one ordering of R ). Since R is Pythagorean, we have already seen that Ann ϕ = ( Q ϕ ). Hence by the theorem � ( X − n ) if s = n , Ann ϕ = (( X − s )( X − n )) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 20 / 32
Formally Real Fields Example Let ϕ be a quadratic form over R with n = dim( ϕ ). Set s := sign( { ϕ } ) (there exists only one ordering of R ). Since R is Pythagorean, we have already seen that Ann ϕ = ( Q ϕ ). Hence by the theorem � ( X − n ) if s = n , Ann ϕ = (( X − s )( X − n )) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 20 / 32
Local Fields Local Fields In this section K is a local field with finite residue field. Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields Local Fields In this section K is a local field with finite residue field. Proposition Let ϕ be a quadratic form over K , n := dim( ϕ ), ϕ �∼ = n × � 1 � . Then a complete and minimal set of generators for Ann ϕ ⊂ Z [ X ] is given as follows: Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields Local Fields In this section K is a local field with finite residue field. Proposition Let ϕ be a quadratic form over K , n := dim( ϕ ), ϕ �∼ = n × � 1 � . Then a complete and minimal set of generators for Ann ϕ ⊂ Z [ X ] is given as follows: if det( ϕ ) is a sum of � 2( X − n ) , ( X − n ) 2 � two squares in K , Ann ϕ = � � 4( X − n ) , ( X − n + 2)( X − n ) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields Local Fields In this section K is a local field with finite residue field. Proposition Let ϕ be a quadratic form over K , n := dim( ϕ ), ϕ �∼ = n × � 1 � . Then a complete and minimal set of generators for Ann ϕ ⊂ Z [ X ] is given as follows: if det( ϕ ) is a sum of � 2( X − n ) , ( X − n ) 2 � two squares in K , Ann ϕ = � � 4( X − n ) , ( X − n + 2)( X − n ) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields Local Fields In this section K is a local field with finite residue field. Proposition Let ϕ be a quadratic form over K , n := dim( ϕ ), ϕ �∼ = n × � 1 � . Then a complete and minimal set of generators for Ann ϕ ⊂ Z [ X ] is given as follows: if det( ϕ ) is a sum of � 2( X − n ) , ( X − n ) 2 � two squares in K , Ann ϕ = � � 4( X − n ) , ( X − n + 2)( X − n ) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields Local Fields In this section K is a local field with finite residue field. Proposition Let ϕ be a quadratic form over K , n := dim( ϕ ), ϕ �∼ = n × � 1 � . Then a complete and minimal set of generators for Ann ϕ ⊂ Z [ X ] is given as follows: if det( ϕ ) is a sum of � 2( X − n ) , ( X − n ) 2 � two squares in K , Ann ϕ = � � 4( X − n ) , ( X − n + 2)( X − n ) otherwise . If ϕ ∼ = n × � 1 � , then Ann ϕ = ( X − n ). Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields Recall Let ϕ ∼ = � a 1 , . . . , a n � be a quadratic form over K . The determinant of ϕ is defined as det( ϕ ) := a 1 · · · a n := a 1 · · · a n ( K ∗ ) 2 ∈ K ∗ / ( K ∗ ) 2 . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 22 / 32
Local Fields Recall Let ϕ ∼ = � a 1 , . . . , a n � be a quadratic form over K . The determinant of ϕ is defined as det( ϕ ) := a 1 · · · a n := a 1 · · · a n ( K ∗ ) 2 ∈ K ∗ / ( K ∗ ) 2 . The Hasse invariant of ϕ is defined as � a i , a j � � s ( ϕ ) := ∈ Br( K ) . K 1 ≤ i < j ≤ n Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 22 / 32
Local Fields Recall Let ϕ ∼ = � a 1 , . . . , a n � be a quadratic form over K . The determinant of ϕ is defined as det( ϕ ) := a 1 · · · a n := a 1 · · · a n ( K ∗ ) 2 ∈ K ∗ / ( K ∗ ) 2 . The Hasse invariant of ϕ is defined as � a i , a j � � s ( ϕ ) := ∈ Br( K ) . K 1 ≤ i < j ≤ n It is well-known that quadratic forms over a local field are classified by dimension, determinant and discriminant. Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 22 / 32
Local Fields Recall Let ϕ ∼ = � a 1 , . . . , a n � be a quadratic form over K . The determinant of ϕ is defined as det( ϕ ) := a 1 · · · a n := a 1 · · · a n ( K ∗ ) 2 ∈ K ∗ / ( K ∗ ) 2 . The Hasse invariant of ϕ is defined as � a i , a j � � s ( ϕ ) := ∈ Br( K ) . K 1 ≤ i < j ≤ n It is well-known that quadratic forms over a local field are classified by dimension, determinant and discriminant. The ideal I ( K ) ⊂ W ( K ) consisting of equivalence classes of even dimensional quadratic forms is called fundamental ideal of W ( K ). Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 22 / 32
Local Fields Proof of the Proposition Idea of Proof. Calculate the Hasse invariants and determinants of ( X − n ) 2 , 2( X − n ) , 4( X − n ) resp. ( X − n + 2)( X − n ) applied to ϕ . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 23 / 32
Local Fields Proof of the Proposition Idea of Proof. Calculate the Hasse invariants and determinants of ( X − n ) 2 , 2( X − n ) , 4( X − n ) resp. ( X − n + 2)( X − n ) applied to ϕ . Then compare these Hasse invariants (resp. determinants) with the Hasse invariants (resp. determinants) of the hyperbolic forms 2 n 2 × H , (2 n 2 + 2 n ) × H . 2 n × H , 4 n × H resp. Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 23 / 32
Local Fields Classification of Annihilating Ideals over Local Fields In the case that the residue field K has characteristic � = 2, we can reformulate the proposition using the valuation v K of K . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 24 / 32
Local Fields Classification of Annihilating Ideals over Local Fields In the case that the residue field K has characteristic � = 2, we can reformulate the proposition using the valuation v K of K . Theorem Let ϕ be a quadratic form over K , n = dim( ϕ ). Assume char( K ) � = 2. (a) If ϕ ∼ = n × � 1 � , then Ann ϕ = ( X − n ). Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 24 / 32
Local Fields Classification of Annihilating Ideals over Local Fields In the case that the residue field K has characteristic � = 2, we can reformulate the proposition using the valuation v K of K . Theorem Let ϕ be a quadratic form over K , n = dim( ϕ ). Assume char( K ) � = 2. (a) If ϕ ∼ = n × � 1 � , then Ann ϕ = ( X − n ). (b) If ϕ �∼ = n × � 1 � , and � 2( X − n ) , ( X − n ) 2 � (i) if − 1 ∈ ( K ∗ ) 2 , then Ann ϕ = . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 24 / 32
Local Fields Classification of Annihilating Ideals over Local Fields In the case that the residue field K has characteristic � = 2, we can reformulate the proposition using the valuation v K of K . Theorem Let ϕ be a quadratic form over K , n = dim( ϕ ). Assume char( K ) � = 2. (a) If ϕ ∼ = n × � 1 � , then Ann ϕ = ( X − n ). (b) If ϕ �∼ = n × � 1 � , and � 2( X − n ) , ( X − n ) 2 � (i) if − 1 ∈ ( K ∗ ) 2 , then Ann ϕ = . (ii) if − 1 �∈ ( K ∗ ) 2 , then �� 2( X − n ) , ( X − n ) 2 � if v K (det( ϕ )) is even , Ann ϕ = � � 4( X − n ) , ( X − n + 2)( X − n ) if v K (det( ϕ )) is odd . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 24 / 32
Global Fields Global Fields In this section K is a global field. Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 25 / 32
Global Fields Global Fields In this section K is a global field. Let V be the set of equivalence classes of absolute values of K . For every ν ∈ V choose a representative | · | ν of the class ν . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 25 / 32
Global Fields Global Fields In this section K is a global field. Let V be the set of equivalence classes of absolute values of K . For every ν ∈ V choose a representative | · | ν of the class ν . Denote by K ν the completion of K with respect to | · | ν . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 25 / 32
Global Fields Global Fields In this section K is a global field. Let V be the set of equivalence classes of absolute values of K . For every ν ∈ V choose a representative | · | ν of the class ν . Denote by K ν the completion of K with respect to | · | ν . We can write V as the disjoint union V = V R ∪ V C ∪ V fin such that R for ν ∈ V R , K ν = C for ν ∈ V C , local for ν ∈ V fin . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 25 / 32
Global Fields First Observations Let ϕ be an n -dimensional quadratic form over K , and let f ∈ Z [ X ]. By the Hasse-Minkowski Theorem f is an annihilating polynomial of ϕ ⇐ ⇒ f is an annihilating polynomial of ϕ K ν for all ν ∈ V . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 26 / 32
Global Fields First Observations Let ϕ be an n -dimensional quadratic form over K , and let f ∈ Z [ X ]. By the Hasse-Minkowski Theorem f is an annihilating polynomial of ϕ ⇐ ⇒ f is an annihilating polynomial of ϕ K ν for all ν ∈ V . Since Ann ϕ K ν = ( X − n ) for all ν ∈ V C , and since X − n divides every annihilating polynomial of ϕ , we do not have to take into account the completions K ν for ν ∈ V C . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 26 / 32
Global Fields First Observations Let ϕ be an n -dimensional quadratic form over K , and let f ∈ Z [ X ]. By the Hasse-Minkowski Theorem f is an annihilating polynomial of ϕ ⇐ ⇒ f is an annihilating polynomial of ϕ K ν for all ν ∈ V . Since Ann ϕ K ν = ( X − n ) for all ν ∈ V C , and since X − n divides every annihilating polynomial of ϕ , we do not have to take into account the completions K ν for ν ∈ V C . Proposition � Ann ϕ = Ann ϕ K ν ν ∈ V R ∪ V fin Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 26 / 32
Global Fields Orderings of Global Fields Assume that V R � = ∅ . = ⇒ K is formally real. Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 27 / 32
Global Fields Orderings of Global Fields Assume that V R � = ∅ . = ⇒ K is formally real. There exists a one-to-one correspondence V R ← → X K between the real completions of K and the space of orderings of K . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 27 / 32
Global Fields Orderings of Global Fields Assume that V R � = ∅ . = ⇒ K is formally real. There exists a one-to-one correspondence V R ← → X K between the real completions of K and the space of orderings of K . More specifically: Every signature homomorphism W ( K ) → Z factors uniquely as Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 27 / 32
Global Fields Orderings of Global Fields Assume that V R � = ∅ . = ⇒ K is formally real. There exists a one-to-one correspondence V R ← → X K between the real completions of K and the space of orderings of K . More specifically: Every signature homomorphism W ( K ) → Z factors uniquely as sign ε ν W ( K ) − − − → W ( R ) − − − → Z , where ε ν is induced by the completion K ֒ → K ν , and sign is the usual signature homomorphism over R . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 27 / 32
Global Fields The Signature Polynomial revisited Recall that S sign = { sign A ( { ϕ } ) | A ∈ X K } ∪ { n } ⊂ Z ϕ and � P sign = ( X − s ) ∈ Z [ X ] . ϕ s ∈ S sign ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 28 / 32
Global Fields The Signature Polynomial revisited Recall that S sign = { sign A ( { ϕ } ) | A ∈ X K } ∪ { n } ⊂ Z ϕ and � P sign = ( X − s ) ∈ Z [ X ] . ϕ s ∈ S sign ϕ By our previous observations we obtain: Lemma The signature polynomial P sign annihilates ϕ K ν for all ν ∈ V R . ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 28 / 32
Global Fields Classification of Annihilating Ideals over Global Fields I Theorem Let ϕ be a quadratic form over K , n = dim( ϕ ). (a) If ϕ ∼ = n × � 1 � , then Ann ϕ = ( X − n ). Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 29 / 32
Global Fields Classification of Annihilating Ideals over Global Fields I Theorem Let ϕ be a quadratic form over K , n = dim( ϕ ). (a) If ϕ ∼ = n × � 1 � , then Ann ϕ = ( X − n ). (b) If ϕ �∼ = n × � 1 � , and (i) if | S sign | = 1, then ϕ if det( ϕ ) is a sum of � 2( X − n ) , ( X − n ) 2 � two squares in K , Ann ϕ = � � 4( X − n ) , ( X − n + 2)( X − n ) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 29 / 32
Global Fields Classification of Annihilating Ideals over Global Fields I Theorem Let ϕ be a quadratic form over K , n = dim( ϕ ). (a) If ϕ ∼ = n × � 1 � , then Ann ϕ = ( X − n ). (b) If ϕ �∼ = n × � 1 � , and (i) if | S sign | = 1, then ϕ if det( ϕ ) is a sum of � 2( X − n ) , ( X − n ) 2 � two squares in K , Ann ϕ = � � 4( X − n ) , ( X − n + 2)( X − n ) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 29 / 32
Global Fields Classification of Annihilating Ideals over Global Fields I Theorem Let ϕ be a quadratic form over K , n = dim( ϕ ). (a) If ϕ ∼ = n × � 1 � , then Ann ϕ = ( X − n ). (b) If ϕ �∼ = n × � 1 � , and (i) if | S sign | = 1, then ϕ if det( ϕ ) is a sum of � 2( X − n ) , ( X − n ) 2 � two squares in K , Ann ϕ = � � 4( X − n ) , ( X − n + 2)( X − n ) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 29 / 32
Global Fields Classification of Annihilating Ideals over Global Fields II Theorem Let ϕ be a quadratic form over K , n = dim( ϕ ). (b) If ϕ �∼ = n × � 1 � , and (ii) if | S sign | = 2 with S sign = { s , n } , then ϕ ϕ if s ≡ n (mod 4) and det( ϕ K ν ) � 2( X − s )( X − n ) , is not a sum of two squares in ( X − s )( X − n ) 2 � Ann ϕ = K ν for some ν ∈ V fin , � � ( X − s )( X − n ) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 30 / 32
Global Fields Classification of Annihilating Ideals over Global Fields II Theorem Let ϕ be a quadratic form over K , n = dim( ϕ ). (b) If ϕ �∼ = n × � 1 � , and (ii) if | S sign | = 2 with S sign = { s , n } , then ϕ ϕ if s ≡ n (mod 4) and det( ϕ K ν ) � 2( X − s )( X − n ) , is not a sum of two squares in ( X − s )( X − n ) 2 � Ann ϕ = K ν for some ν ∈ V fin , � � ( X − s )( X − n ) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 30 / 32
Global Fields Classification of Annihilating Ideals over Global Fields II Theorem Let ϕ be a quadratic form over K , n = dim( ϕ ). (b) If ϕ �∼ = n × � 1 � , and (ii) if | S sign | = 2 with S sign = { s , n } , then ϕ ϕ if s ≡ n (mod 4) and det( ϕ K ν ) � 2( X − s )( X − n ) , is not a sum of two squares in ( X − s )( X − n ) 2 � Ann ϕ = K ν for some ν ∈ V fin , � � ( X − s )( X − n ) otherwise . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 30 / 32
Global Fields Classification of Annihilating Ideals over Global Fields II Theorem Let ϕ be a quadratic form over K , n = dim( ϕ ). (b) If ϕ �∼ = n × � 1 � , and (ii) if | S sign | = 2 with S sign = { s , n } , then ϕ ϕ if s ≡ n (mod 4) and det( ϕ K ν ) � 2( X − s )( X − n ) , is not a sum of two squares in ( X − s )( X − n ) 2 � Ann ϕ = K ν for some ν ∈ V fin , � � ( X − s )( X − n ) otherwise . (iii) if | S sign | ≥ 3, then Ann ϕ = ( P sign ). ϕ ϕ Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 30 / 32
Global Fields Proof Sketch of Proof. The claims (a) and (b).(iii) follow directly from our previous observations. Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 31 / 32
Global Fields Proof Sketch of Proof. The claims (a) and (b).(iii) follow directly from our previous observations. For the proof of (b).(i) one shows that one does not have to take into account the real completions K ν for ν ∈ V R . Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 31 / 32
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