1. Galois Theory and Phil Pollard

2. To Be Spoken: The purpose of this project is to present a brief introduction to Galois Theory and to illustrate it’s uses in focusing infinite situations into a finite workspace. These slides will focus mainly on the use of Galois Theory to prove the insolvability of the quintic polynomial by a general equation using radicals. Most of the descriptions of theorems and properties are paraphrased interpretations rather than rigorous proofs.

3. Things you should know by now  Permutation Groups  Extension Fields  Splitting Fields (And if you don’t, you won’t know what the hell I’m talking about.)

4. Etymology  F := An arbitrary Field (assume char = 0)  E := An extension Field over F  ℚ := Field of Rational Numbers  F[x] := Polynomial Ring over F  F(a 1,a 2,…) := Extension of F by a 1, a 2,…  e := Identity (of current discussed group)  p.g. := “Proof Gist” expedited proof. (Unless otherwise stated)

5. Review of Terms Degree of Extension Automorphism Denoted [E:F] = n Where n is the dimension of E as a vector space over F. A ring isomorphism from a Field onto itself Simple Group A group with no proper, non-trivial normal subgroup Field of Rational Functions Denoted F(x 1, x 2,…,x n ) = { | f,g ∈ F[x 1, x 2,…,x n ] } f(x 1, x 2,…,x n ) g(x 1, x 2,…,x n )

6. New Definitions Galois Group Denoted Gal(E/F) [read Galois Group of E fixing F] The set of all automorphisms of E that “fix” F (take each element of F to itself) The Galois Group is a group under function composition. Fixed Field Denoted E H for H ⊆ Gal(E/F) E H = {x ∈ E | Φ(x)=x ∀ Φ ∈ H} Note: E Gal(E/F) = F Intermediate Field F ≤ K ≤ E and K is a field then K is an intermediate field and Ģ(K) := Gal(E/K) Note: E Ģ(K) = K

7. Theorem 1 If f(x) is irreducible in F[x] and E is a splitting field of f(x), Φ ∈ Gal(E/F) permutes the roots of f(x) p.g. If a i is a root of f(x), then f(a i ) = Φ (f(a i )) = 0because 0 is fixed by Φ = f(Φ(a i ))Operation Preserving But since a i ∉ F, it is not necessarily fixed Obvious Corollary 1 If f(x) ∈ F[x] has degree n, then Gal(E/F) is isomorphic to a subgroup of S n (E being a splitting field of F)

8. Example 1: Consider E = ℚ (ω, 3 √2) where ω = -1+i√3 2 Then the elements of Gal(E/ ℚ ) are described as follows: eαββ2β2 αβαβ 2 ω→ωω→ωω→ω2ω→ω2 ω→ωω→ωω→ωω→ωω→ω2ω→ω2 ω→ω2ω→ω2 3 √2→ 3 √2 3 √2→ω · 3 √2 3 √2→ω 2 · 3 √2 3 √2→ω · 3 √2 Note: Gal(E/ ℚ ) is not Abelian. αβ ≠ βα

9. Lattices {e, α, β, β 2, αβ, αβ 2 } {e, β, β 2 }{e, α}{e, αβ}{e, αβ 2 } 23332333 {e} 322 2 E = ℚ (ω, 3 √2) ℚ (ω) ℚ ( 3 √2) ℚ (ω · 3 √2) ℚ (ω 2 · 3 √2) ℚ 3 222 23332333 Subgroups of Gal(E/ ℚ ) Subfields of E

10. Fundamental Theorem of Galois Theory (Para. Gallien) Given E, a splitting field over F, F being finite or with character 0 For every subfield K of E containing F There is a subgroup Gal(E/K) of Gal(E/F), and 1.[E:K] = |Gal(E/K)| and [K:F] = |Gal(E/F)| / |Gal(E/K)| 2.If K is a splitting field of some f(x) ∈ F[x] then Gal(E/K) ◁ Gal(E/F) and |Gal(K/F)| = |Gal(E/F)| / |Gal(E/K)| 3.K = E Gal(E/K) 4.For H ≤ Gal(E/F), H = Gal(E/E H ) Proof by Smart Dead Guy. ■ (not para. Gallien)

11. Solvable by Radicals Pure Extension An extension F(a)/F where a m ∈ F (referred to as “type m”) Radical Extension An extension E/F such that F = K 0 ⊆ K 1 ⊆ ··· ⊆ K n-1 ⊆ K n = E creates a tower of fields in which each extension K i+1 /K i is pure Solvable by Radicals f(x) ∈ F[x] is Solvable by Radicals if it has a splitting field E such that E/F forms a radical extension.

12. Solvable Group A group G that has a series of normal subgroups {e} = H 0 ◁ H 1 ◁ ··· ◁ H n-1 ◁ H n = G where for each 0 ≤ i ≤ n, H i+1 /H i is Abelian Note 1: The following are easy to show: · Abelian Groups are solvable · Non-Abelian Simple Groups are not solvable · Subgroups of solvable groups are also solvable (slightly less easy) Note 2: We will eventually need the following Theorem: Theorem 2 For E, a splitting field for x n – a over F, Gal(E/F) is solvable. Proof by Smart Dead Guy ■

13. Suspiciously Convenient Example S 5 is not a solvable group p.g. Through an exhaustive orders argument, we find that A 5 is a simple group. This means that A 5 itself is not a solvable group. And since subgroups of solvable groups are solvable, S 5 and S n for n ≥ 5 cannot be solvable. ■ (The viewer is spared the details, but a full proof of the simplicity of A 5 can be found online or in almost any Algebra text.)

14. Solvable by Radicals implies Solvable group For E = F(a 1,…,a n ), a Radical Extension for f(x) ∈ F[x], Gal(E/F) is solvable This can be shown by induction. For the Base Case, E = F(a 1 ), E splits some function f(x) = x n – a, and Gal(E/F) is therefore solvable by Theorem 2. For the complete induction proof, the viewer is referred to Gallien, page 556. Proof by Smart Dead Guy ■ Obvious Contraposition 1 If Gal(E/F) is not solvable, then f(x) is not solvable by radicals.

15. The Galois Group for Certain Polynomials of n th power (para. Herstein) Consider F(x 1,…,x n ) and let S be the field of symmetric rational functions over F. Consider the polynomial: f(y) = y n – a 1 y n-1 + a 2 y n-2 – ··· + (-1) n a n in S[y] with a 1,…,a n ∈ S The roots of f(y) are not contained in S because a 1,…,a n must have independent, permutable variables But we find that F(x 1,…,x n ) is a splitting field of f(x) over S and f(y) = y n – a 1 y n-1 + a 2 y n-2 – ··· + (-1) n a n = (y – t 1 )(y – t 2 )···(y – t n ) ∈ F(x 1,…,x n ) [y] (t i in terms of x’s) And the roots t 1, …,t n are distinct. Thus, |Gal(F(x 1,…,x n ) / S)| = n! And by Obvious Corollary 1, Gal( F(x 1,…,x n ) / S) is isomorphic to a subgroup of S n Hence, Gal(F(x 1,…,x n ) / S) is isomorphic to S n

16. The Insolvability of the Quintic by Radicals We just found out that certain n-power Functions produce a Galois Group isomorphic to S n And we know by our Suspiciously Convenient Example that S n is not solvable for n ≥ 5 And we know by our Obvious Contraposition that if a Galois Group Gal(E/F) is not a solvable group, the equation split by E is not solvable by radicals. Thus, the Quintic Polynomial is not generally solvable by radicals. Fin.

17. References Contemporary Abstract Algebra, 7 th Edition, Joseph Gallian. 2010, Brooks/Cole Advanced Modern Algebra, Joseph J. Rotman. 2003, Prentice Hall Galois Theory, 2 nd Edition, Ian Stewart. 1989, Chapman & Hall/CRC Topics in Algebra, I.N. Herstein. 1965, Blaisdell Publishing Chapter 6 Notes – Galois Theory, University of Illinois, Urbana-Champaign http://www.math.uiuc.edu/~r-ash/Algebra/Chapter6.pdf