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Constructive Galois Theory with Linear Algebraic Groups

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Constructive Galois Theory with Linear Algebraic Groups Constructive Galois Theory with Linear Algebraic Groups Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales LSU REU, Summer 2015 April 14, 2018 Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups If K ⊆ L is a field extension obtained by adjoining all roots of a family of polynomials*, then L=K is Galois, and Gal(L=K) := Aut(L=K) Idea : Families of polynomials ! L=K ! Gal(L=K) Primer on Galois Theory Let K be a field (e.g., Q; Fq; Fq(t)..). Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Idea : Families of polynomials ! L=K ! Gal(L=K) Primer on Galois Theory Let K be a field (e.g., Q; Fq; Fq(t)..). If K ⊆ L is a field extension obtained by adjoining all roots of a family of polynomials*, then L=K is Galois, and Gal(L=K) := Aut(L=K) Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Primer on Galois Theory Let K be a field (e.g., Q; Fq; Fq(t)..). If K ⊆ L is a field extension obtained by adjoining all roots of a family of polynomials*, then L=K is Galois, and Gal(L=K) := Aut(L=K) Idea : Families of polynomials ! L=K ! Gal(L=K) Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Given a finite group G Does there exist field extensions L=K such that Gal(L=K) =∼ G? Inverse Galois Problem Given Galois field extensions L=K Compute Gal(L=K) Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Inverse Galois Problem Given Galois field extensions L=K Compute Gal(L=K) Given a finite group G Does there exist field extensions L=K such that Gal(L=K) =∼ G? Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 1 \Describe" the splitting polynomial f of a given base field ∼ (say K ⊃ Fq) with Gal(f =K) = G? 2 \Describe" all field extensions E of a given base field (say ∼ K ⊃ Fq) with Gal(E=K) = G? One approach: generic polynomials and generic extensions. Constructive Galois Theory Let G be a finite group. Can we... Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 2 \Describe" all field extensions E of a given base field (say ∼ K ⊃ Fq) with Gal(E=K) = G? One approach: generic polynomials and generic extensions. Constructive Galois Theory Let G be a finite group. Can we... 1 \Describe" the splitting polynomial f of a given base field ∼ (say K ⊃ Fq) with Gal(f =K) = G? Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups One approach: generic polynomials and generic extensions. Constructive Galois Theory Let G be a finite group. Can we... 1 \Describe" the splitting polynomial f of a given base field ∼ (say K ⊃ Fq) with Gal(f =K) = G? 2 \Describe" all field extensions E of a given base field (say ∼ K ⊃ Fq) with Gal(E=K) = G? Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Constructive Galois Theory Let G be a finite group. Can we... 1 \Describe" the splitting polynomial f of a given base field ∼ (say K ⊃ Fq) with Gal(f =K) = G? 2 \Describe" all field extensions E of a given base field (say ∼ K ⊃ Fq) with Gal(E=K) = G? One approach: generic polynomials and generic extensions. Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups ∼ 1 Gal(f (X ; t1; :::; tn)=K(t1; :::; tn)) = G. 2 Every Galois G-extension M=L, where K ⊂ L, is the splitting field of a specialization f (X ; ξ) for some ξ 2 Ln. 2 Example: X − t 2 Q(t)[X ] is Z=2Z-generic. 2 X + t1X + t0 2 Q(t1; t0)[X ] is also Z=2Z-generic, but less efficient. Generic Polynomials Definition A monic separable polynomial f (X ; t1; :::; tn) 2 K(t1; :::; tn)[X ] is called G-generic over K if the following conditions are satisfied: Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 2 Every Galois G-extension M=L, where K ⊂ L, is the splitting field of a specialization f (X ; ξ) for some ξ 2 Ln. 2 Example: X − t 2 Q(t)[X ] is Z=2Z-generic. 2 X + t1X + t0 2 Q(t1; t0)[X ] is also Z=2Z-generic, but less efficient. Generic Polynomials Definition A monic separable polynomial f (X ; t1; :::; tn) 2 K(t1; :::; tn)[X ] is called G-generic over K if the following conditions are satisfied: ∼ 1 Gal(f (X ; t1; :::; tn)=K(t1; :::; tn)) = G. Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 2 Example: X − t 2 Q(t)[X ] is Z=2Z-generic. 2 X + t1X + t0 2 Q(t1; t0)[X ] is also Z=2Z-generic, but less efficient. Generic Polynomials Definition A monic separable polynomial f (X ; t1; :::; tn) 2 K(t1; :::; tn)[X ] is called G-generic over K if the following conditions are satisfied: ∼ 1 Gal(f (X ; t1; :::; tn)=K(t1; :::; tn)) = G. 2 Every Galois G-extension M=L, where K ⊂ L, is the splitting field of a specialization f (X ; ξ) for some ξ 2 Ln. Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 2 X + t1X + t0 2 Q(t1; t0)[X ] is also Z=2Z-generic, but less efficient. Generic Polynomials Definition A monic separable polynomial f (X ; t1; :::; tn) 2 K(t1; :::; tn)[X ] is called G-generic over K if the following conditions are satisfied: ∼ 1 Gal(f (X ; t1; :::; tn)=K(t1; :::; tn)) = G. 2 Every Galois G-extension M=L, where K ⊂ L, is the splitting field of a specialization f (X ; ξ) for some ξ 2 Ln. 2 Example: X − t 2 Q(t)[X ] is Z=2Z-generic. Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Generic Polynomials Definition A monic separable polynomial f (X ; t1; :::; tn) 2 K(t1; :::; tn)[X ] is called G-generic over K if the following conditions are satisfied: ∼ 1 Gal(f (X ; t1; :::; tn)=K(t1; :::; tn)) = G. 2 Every Galois G-extension M=L, where K ⊂ L, is the splitting field of a specialization f (X ; ξ) for some ξ 2 Ln. 2 Example: X − t 2 Q(t)[X ] is Z=2Z-generic. 2 X + t1X + t0 2 Q(t1; t0)[X ] is also Z=2Z-generic, but less efficient. Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Definition Let K be a field and G a finite group. A Galois ring extension S=R with group G is called a generic G-extension over K, if 1 R = K[t1;:::; tn][1=g] 2 whenever L is a field extension of K and M=L a Galois G-algebra, there is a homomorphism of K-algebras ' : R ! L, such that S ⊗' L=L and M=L are isomorphic as Galois extensions. This definition is due to Saltman [6]. Generic Extensions More ambitiously... Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 1 R = K[t1;:::; tn][1=g] 2 whenever L is a field extension of K and M=L a Galois G-algebra, there is a homomorphism of K-algebras ' : R ! L, such that S ⊗' L=L and M=L are isomorphic as Galois extensions. This definition is due to Saltman [6]. Generic Extensions More ambitiously... Definition Let K be a field and G a finite group. A Galois ring extension S=R with group G is called a generic G-extension over K, if Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 2 whenever L is a field extension of K and M=L a Galois G-algebra, there is a homomorphism of K-algebras ' : R ! L, such that S ⊗' L=L and M=L are isomorphic as Galois extensions. This definition is due to Saltman [6]. Generic Extensions More ambitiously... Definition Let K be a field and G a finite group. A Galois ring extension S=R with group G is called a generic G-extension over K, if 1 R = K[t1;:::; tn][1=g] Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Generic Extensions More ambitiously... Definition Let K be a field and G a finite group. A Galois ring extension S=R with group G is called a generic G-extension over K, if 1 R = K[t1;:::; tn][1=g] 2 whenever L is a field extension of K and M=L a Galois G-algebra, there is a homomorphism of K-algebras ' : R ! L, such that S ⊗' L=L and M=L are isomorphic as Galois extensions. This definition is due to Saltman [6]. Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 1 Assuming G satisfies some kind of Hilbert Theorem 90, we explicitly construct generic extensions and generic polynomials for G(Fq).
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