The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions On Klein’s icosahedral solution of the quintic (Irish Mathematics Students Assoc. Conf.) Oliver Nash March 2, 2013 Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions Table of contents The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions General quintic equation This is what we are going to solve for x: x 5 + a 1 x 4 + a 2 x 3 + a 3 x 2 + a 4 x + a 5 = 0 where a 1 , a 2 , . . . , a 5 are any complex numbers. Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions General quintic equation This is what we are going to solve for x: x 5 + a 1 x 4 + a 2 x 3 + a 3 x 2 + a 4 x + a 5 = 0 where a 1 , a 2 , . . . , a 5 are any complex numbers. What do we mean by solve? Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions Solutions of polynomials in radicals ◮ Solving an equation in radicals means using only + , − , · , / and root extraction (finitely many times). ◮ We can solve quadratic, cubic and quartic equations in radicals. Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions Solutions of polynomials in radicals ◮ Solving an equation in radicals means using only + , − , · , / and root extraction (finitely many times). ◮ We can solve quadratic, cubic and quartic equations in radicals. ◮ Famously, we cannot solve the quintic equation in radicals (Abel & Ruffini : 1823, 1799). ◮ Galois developed the most important tools: Galois group permutes roots. S n in general. ◮ We must explore other means of solving the quintic. Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions Tschirnhaus reduction of the quintic ◮ General quintic: x 5 + a 1 x 4 + a 2 x 3 + a 3 x 2 + a 4 x + a 5 = 0. ◮ Well known substitution y = x − a 1 / 5 eliminates degree 4 term. Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions Tschirnhaus reduction of the quintic ◮ General quintic: x 5 + a 1 x 4 + a 2 x 3 + a 3 x 2 + a 4 x + a 5 = 0. ◮ Well known substitution y = x − a 1 / 5 eliminates degree 4 term. ◮ In fact if we allow y to be quadratic in x we can also eliminate degree 3 term. ◮ Thus reduce general quintic to canonical form : y 5 + 5 α y 2 + 5 β y + γ = 0. Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions Geometry of the roots ◮ Vector of roots of quintic is a point in C 5 and so defines point in (complex) projective space P 4 . ◮ If quintic is in canonical form then roots satisfy � y i = � y 2 i = 0 and so point in P 4 lies on quadric surface. ◮ Quadric surface is doubly-ruled surface. This might be useful so let’s remind ourselves what this means. Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions The quadric surface looks like this Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions What are they? ◮ A quadric surface is a 2-dimensional space that is solution of a quadratic equation. ◮ ax 2 + by 2 + cz 2 + dxy + eyz + fzx + gx + hy + iz + j = 0 Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions What are they? ◮ A quadric surface is a 2-dimensional space that is solution of a quadratic equation. ◮ ax 2 + by 2 + cz 2 + dxy + eyz + fzx + gx + hy + iz + j = 0 ◮ Over R these are ellipsoids, paraboloids, hyperboloids (if non-singular). ◮ Much simpler over C and in projective case: all non-singular quadric surfaces can be put in form XY = ZW for homogeneous coordinates [ X , Y , Z , W ] for P 3 . Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions Double ruling explicitly Given: XY = ZW } ⊂ P 3 Q = { [ X , Y , Z , W ] | We have bijection: Q → P 1 × P 1 [ X , Y , Z , W ] �→ ([ X , Z ] , [ X , W ]) with inverse: ([ λ 0 , λ 1 ] , [ µ 0 , µ 1 ]) �→ [ λ 0 µ 0 , λ 1 µ 1 , λ 1 µ 0 , λ 0 µ 1 ] Fibres over each factor are lines in Q Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions A family of lines Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions The complex projective line is a sphere The projective line adds just one point, namely ∞ , to the affine line: P 1 = C ∪ {∞} . In homogeneous coordinates ∞ = [1 , 0]. Stereographic projection gives natural bijection S 2 → C ∪ {∞} thus have natural bijection S 2 → P 1 . Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions The complex projective line is a sphere The projective line adds just one point, namely ∞ , to the affine line: P 1 = C ∪ {∞} . In homogeneous coordinates ∞ = [1 , 0]. Stereographic projection gives natural bijection S 2 → C ∪ {∞} thus have natural bijection S 2 → P 1 . For sake of definiteness, we give coordinate representation of S 2 → P 1 : ( x , y , z ) �→ [ x + iy , 1 − z ] with inverse: 1 uv ) , | u | 2 − | v | 2 ) [ u , v ] �→ | u | 2 + | v | 2 ( u ¯ v + ¯ uv , − i ( u ¯ v − ¯ Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions Stereographic projection looks like this Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
The quintic equation Quadric surfaces The complex projective line Action of the Galois group The icosahedral map Putting it all together Career opportunities at SIG Questions Roots not naturally ordered ◮ Roots unordered so actually just get S 5 -orbit on quadric surface. ◮ S 5 action is just permutation of coordinates so is linear and so is induced from action on the two families of lines in Q . ◮ Odd permutations interchange two families so restrict to A 5 . Oliver Nash On Klein’s icosahedral solution of the quintic (Irish Mathematics
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