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

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 ∈ Ln.

2 Example: X − t ∈ Q(t)[X ] is Z/2Z-generic. 2 X + t1X + t0 ∈ Q(t1, t0)[X ] is also Z/2Z-generic, but less eﬃcient.

Generic Polynomials

Deﬁnition

A monic separable polynomial f (X ; t1, ..., tn) ∈ 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 ∈ Ln.

2 Example: X − t ∈ Q(t)[X ] is Z/2Z-generic. 2 X + t1X + t0 ∈ Q(t1, t0)[X ] is also Z/2Z-generic, but less eﬃcient.

Generic Polynomials

Deﬁnition

A monic separable polynomial f (X ; t1, ..., tn) ∈ 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 ∈ Q(t)[X ] is Z/2Z-generic. 2 X + t1X + t0 ∈ Q(t1, t0)[X ] is also Z/2Z-generic, but less eﬃcient.

Generic Polynomials

Deﬁnition

A monic separable polynomial f (X ; t1, ..., tn) ∈ 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 ∈ Ln.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 2 X + t1X + t0 ∈ Q(t1, t0)[X ] is also Z/2Z-generic, but less eﬃcient.

Generic Polynomials

Deﬁnition

2 Example: X − t ∈ 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

2 Example: X − t ∈ Q(t)[X ] is Z/2Z-generic. 2 X + t1X + t0 ∈ 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 ϕ : R → L, such that S ⊗ϕ 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 ϕ : R → L, such that S ⊗ϕ 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 ϕ : R → L, such that S ⊗ϕ 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

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). 2 We apply this construction to selected algebraic groups to deduce similar results for other ﬁnite groups.

Summary of Results

n Let K be a ﬁeld containing Fq, where q = p , and G an algebraic group over Fq.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 2 We apply this construction to selected algebraic groups to deduce similar results for other ﬁnite groups.

Summary of Results

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Summary of Results

n Let K be a field containing Fq, where q = p , and G an algebraic group over Fq. 1 Assuming G satisfies some kind of Hilbert Theorem 90, we explicitly construct generic extensions and generic polynomials for G(Fq). 2 We apply this construction to selected algebraic groups to deduce similar results for other finite groups.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups ∗ Then we have an induced map λ : Fq[G] → Fq[G] deﬁned by

f 7→ f ◦ λ

The Lang Map

Let λ : G → G be the Lang map deﬁned by

X 7→ X σ(X )−1

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups The Lang Map

Let λ : G → G be the Lang map deﬁned by

X 7→ X σ(X )−1

∗ Then we have an induced map λ : Fq[G] → Fq[G] deﬁned by

f 7→ f ◦ λ

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 1 2 Let L be a ﬁeld containing Fq such that H (L, G) = {1}, and let M/L be a Galois G(Fq)-algebra. Then there exists an ∼ Fq-algebra homomorphism ϕ : R → L such that L ⊗ϕ S = M as G(Fq)-algebras over L.

Main Theorem

Theorem. [C.-Ferrara-Mazurowski]

1 Let λ : G → G be the Lang map. Let S = Fq[G] and ∗ R = λ (Fq[G]). Then S/R is a Galois extension of rings with group G(Fq).

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Main Theorem

Theorem. [C.-Ferrara-Mazurowski]

1 Let λ : G → G be the Lang map. Let S = Fq[G] and ∗ R = λ (Fq[G]). Then S/R is a Galois extension of rings with group G(Fq). 1 2 Let L be a ﬁeld containing Fq such that H (L, G) = {1}, and let M/L be a Galois G(Fq)-algebra. Then there exists an ∼ Fq-algebra homomorphism ϕ : R → L such that L ⊗ϕ S = M as G(Fq)-algebras over L.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups We focus only on closed, connected unipotent groups. Theorem (Hilbert’s Theorem 90 for Unipotent Groups) Let U be a unipotent group deﬁned over a perfect ﬁeld K, and L an extension of K. Then H1(L, U) is trivial.

Application I: Unipotent Groups

Deﬁnition

A unipotent group U is a subgroup of GLn that is conjugate to a subgroup of

1 ∗ · · · ∗     0 1 ∗ · · ·  Un =  .  ∈ GLn  ..  0 0 ∗    0 0 0 1 

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Theorem (Hilbert’s Theorem 90 for Unipotent Groups) Let U be a unipotent group deﬁned over a perfect ﬁeld K, and L an extension of K. Then H1(L, U) is trivial.

Application I: Unipotent Groups

Deﬁnition

A unipotent group U is a subgroup of GLn that is conjugate to a subgroup of

1 ∗ · · · ∗     0 1 ∗ · · ·  Un =  .  ∈ GLn  ..  0 0 ∗    0 0 0 1 

We focus only on closed, connected unipotent groups.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Application I: Unipotent Groups

Deﬁnition

A unipotent group U is a subgroup of GLn that is conjugate to a subgroup of

1 ∗ · · · ∗     0 1 ∗ · · ·  Un =  .  ∈ GLn  ..  0 0 ∗    0 0 0 1 

We focus only on closed, connected unipotent groups. Theorem (Hilbert’s Theorem 90 for Unipotent Groups) Let U be a unipotent group deﬁned over a perfect ﬁeld K, and L an extension of K. Then H1(L, U) is trivial.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Example: Let Q = hi, j| : i 2 = j2 = (ij)2 = −1i be the quaternion group. We can realize Q as a subgroup of U4(F2) via 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 i 7→   , j 7→   0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1

Application I: Unipotent Groups

We can immediately deduce: Theorem. [C.-Ferrara-Mazurowski] Let U be a closed connected unipotent group, and q = pm a prime power. Then there exists a U(Fq)-generic polynomial in characteristic p.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups We can realize Q as a subgroup of U4(F2) via 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 i 7→   , j 7→   0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1

Application I: Unipotent Groups

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Application I: Unipotent Groups

We can immediately deduce: Theorem. [C.-Ferrara-Mazurowski] Let U be a closed connected unipotent group, and q = pm a prime power. Then there exists a U(Fq)-generic polynomial in characteristic p. Example: Let Q = hi, j| : i 2 = j2 = (ij)2 = −1i be the quaternion group. We can realize Q as a subgroup of U4(F2) via 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 i 7→   , j 7→   0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups n−1 qn X (qn−1−qi−1) qi n (qn−1−1) f (X ) = X + ti s X + (−1) s X i=1

is generic for SLn(Fq) over Fq.

Other applications: split solvable groups, certain split semisimple groups, tori.

Examples

Examples:

f (X ) = X 8 + (s2 + st + t2 + 1)X 4 + ··· + s2t4 + 3t6

is generic for Q8 over F2.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Other applications: split solvable groups, certain split semisimple groups, tori.

Examples

Examples:

f (X ) = X 8 + (s2 + st + t2 + 1)X 4 + ··· + s2t4 + 3t6

is generic for Q8 over F2.

n−1 qn X (qn−1−qi−1) qi n (qn−1−1) f (X ) = X + ti s X + (−1) s X i=1

is generic for SLn(Fq) over Fq.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Examples

Examples:

f (X ) = X 8 + (s2 + st + t2 + 1)X 4 + ··· + s2t4 + 3t6

is generic for Q8 over F2.

n−1 qn X (qn−1−qi−1) qi n (qn−1−1) f (X ) = X + ti s X + (−1) s X i=1

is generic for SLn(Fq) over Fq.

Other applications: split solvable groups, certain split semisimple groups, tori.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Example:

f (X ) = X 8 + (4s3 + 2st2)X 4 + s2t4 + 3t6

is generic for Z/8Z over F5.

Application III: Tori and Cyclic 2-Groups

Theorem. [C.-Ferrara-Mazurowski] m There exists a Z/2 Z generic polynomial in characteristic p > 2 (with minimal parameters).

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Application III: Tori and Cyclic 2-Groups

Theorem. [C.-Ferrara-Mazurowski] m There exists a Z/2 Z generic polynomial in characteristic p > 2 (with minimal parameters). Example:

f (X ) = X 8 + (4s3 + 2st2)X 4 + s2t4 + 3t6

is generic for Z/8Z over F5.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 1 Generic polynomials exist for all odd cyclic groups over any characteristic. 2 If G and H have generic polynomials, so does G × H.

Corollary. [C.-Ferrara-Mazurowski] Let p be an odd prime. There exists Z/nZ-generic polynomials in characteristic p for any n. Contrast with Theorem. [Lenstra] Let G be a ﬁnite cyclic group. Then G-generic polynomials exist over Q if and only if G has no elements of order 8.

Application III: Tori and Cyclic 2-Groups

Previously known results:

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups 2 If G and H have generic polynomials, so does G × H.

Application III: Tori and Cyclic 2-Groups

Previously known results: 1 Generic polynomials exist for all odd cyclic groups over any characteristic.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Contrast with Theorem. [Lenstra] Let G be a ﬁnite cyclic group. Then G-generic polynomials exist over Q if and only if G has no elements of order 8.

Application III: Tori and Cyclic 2-Groups

Previously known results: 1 Generic polynomials exist for all odd cyclic groups over any characteristic. 2 If G and H have generic polynomials, so does G × H.

Corollary. [C.-Ferrara-Mazurowski] Let p be an odd prime. There exists Z/nZ-generic polynomials in characteristic p for any n.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups Application III: Tori and Cyclic 2-Groups

Previously known results: 1 Generic polynomials exist for all odd cyclic groups over any characteristic. 2 If G and H have generic polynomials, so does G × H.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups ReferencesI

M. Albert and A. Maier, Additive polynomials for finite groups of lie type, Israel Journal of Mathematics 186 (2011), 125–195. J. Buhler and Z. Reichstein, On the essential dimension of a finite group, Composito Mathematica 106 (1997), 159–179. N. Elkies, Linearized algebra and finite groups of lie type: I: Linear and symplectic groups, Contemporary Mathematics 245 (1999), 82. D. Garling, A course in galois theory, Cambridge University Press, 1986. J. Humphreys, Linear algebraic groups, Springer, 1981.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups ReferencesII

C. Jensen, A. Ledet, and N. Yui, Generic polynomials: Constructive aspects of the inverse galois problem, vol. 45, Cambridge University Press, 2002. N. Karpenko and A. Merkurjev, Essential dimension of finite p-groups, Inventiones mathematicae 172 (2008), 491–508. G. Kemper and E. Mattig, Generic polynomials with few paramters, J. Symbolic Computation 30 (2000), 843–857. M. Knus, A. Merkurjev, M. Rost, and J. Tignol, The book of involutions, Colloquium Publications, American Mathematical Society, 1998. S. Lang, Algebraic groups over finite fields, American Journal of Mathematics 78 (1956), no. 3, 555–563.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups ReferencesIII

Heinrich Matzat, Frobenius modules and galois groups, Kluwer Academic Publishers, 2003. J. Morales and A. Sanchez, Generic extensions and generic polynomials for multiplicative groups, Journal of Algebra 423 (2014), 405–421. G. Prasad and A. Rapinchuk, Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, vol. 109, Publications mathematiques, Jul. 2009. D. Saltman, Generic galois extensions and problems in ﬁeld theory, Advances in Mathematics 43 (1982), 250–283. R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathmatical Society 80 (1968).

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups ReferencesIV

W.C. Waterhouse, Introduction to aﬃne group schemes, Springer, 1979.

Eric Chen, J.T. Ferrara, Liam Mazurowski, Prof. Jorge Morales Constructive Galois Theory with Linear Algebraic Groups