EXPLICIT REALIZATION OF THE DICKSON GROUPS G_{2}(q) AS GALOIS GROUPS

GUNTER MALLE

Abstract. For any prime power q we determine a polynomial f_{q}(X) ∈ 𝔽_{q}(t, u)[X] whose Galois group over 𝔽_{q}(t, u) is the Dickson group G_{2}(q). The construction uses a criterion and a method due to Matzat.

1. Introduction

In this paper we are concerned with the construction of polynomials whose Galois groups are the exceptional simple Chevalley groups G_{2}(q), q a prime power, first discovered by Dickson; see Theorems 4.1 and 4.3.

It was shown by Nori [7] that all semisimple simply-connected linear algebraic groups over 𝔽_{q} do occur as Galois groups of regular extension of function fields over 𝔽_{q}, but his proof does not give an explicit equation or even a constructive method for obtaining such extensions. On the other hand, in a long series of papers Abhyankar has given families of polynomials for groups of classical types (see [1] and the references cited there). His ad hoc approach hasn’t yet led to families with groups of exceptional type (but see [2] for a different construction of polynomials with Galois group the simple groups of Suzuki). Thus it seems natural to try to fill this gap. In his recent paper Matzat [6] describes an algorithmic approach which reduces the construction of generating polynomials for such extensions to certain group theoretic calculations.

More precisely, let F := 𝔽_{q}(t), with t = (t_{1}, …, t_{s}) a set of indeterminates. We denote by ϕ_{q} : F → F, x ↦ x^{q}, the Frobenius endomorphism. Let G be a reduced connected linear algebraic group defined over 𝔽_{q}, with a faithful linear representation Γ : G(F) ↪ GL_{n}(F) in its defining characteristic, also defined over 𝔽_{q}. We identify G(F) with its image in GL_{n}(F). Fix an element g ∈ G(F) and assume that g ∈ GL_{n}(R), where R := 𝔽_{q}[t]. Any specialization homomorphism ψ : R → 𝔽_{q^{a}}, t_{j} ↦ ψ(t_{j}), can be naturally extended to GL_{n}(R). We define

g_{ψ} := ψ(g) · ψ(ϕ_{q}(g)) ⋯ ψ(ϕ_{q}^{a−1}(g)) ∈ GL_{n}(𝔽_{q}).

With these notations Matzat [6, Thm. 4.3 and 4.5] shows the following:

Theorem 1.1. (Matzat) Let G(F) ≤ GL_{n}(F) be a reduced connected linear algebraic group defined over 𝔽_{q}. Let g ∈ GL_{n}(R) such that

(i) g ∈ G(F), (ii) there exist specializations ψ_{i} : R → 𝔽_{q^{a_{i}}}, 1 ≤ i ≤ k, such that no proper

subgroup of G(𝔽_{q}) ≤ GL_{n}(𝔽_{q}) contains conjugates of all the g_{ψ_{i}}, 1 ≤ i ≤ k.

1991 Mathematics Subject Classification. 12F12.

Then G(𝔽_{q}) occurs as regular Galois group over F, and a generating polynomial f(t, X) ∈ F [X] for such a G(𝔽_{q})-extension can be computed explicitly from the matrix g.

Thus the strategy for the computation for a G_{2}(q)-polynomial will be the following. First construct a small faithful matrix representation of G_{2}(F) in its defining characteristic. For this we use the well-known facts that G_{2}(F) is a subgroup of an 8-dimensional orthogonal group over F, and that this 8-dimensional representation has a faithful irreducible constituent of dimension 6 for G_{2}(F), if char(F) = 2 respectively of dimension 7 if char(F) > 2. Secondly, we need to find an element g ∈ G_{2}(F) with the properties required in the Theorem. For this, we make use of the known lists of maximal subgroups of G_{2}(q) by Cooperstein and Kleidman. (These results require the classification of finite simple groups, but only in a very weak form.) Finally, the corresponding polynomial has to be computed using a version of the Buchberger algorithm. Acknowledgement: I’m indebted to N. Elkies for pointing out an overzealous simplification in a previous version.

2. Identifying G_{2}(F) inside the 8-dimensional orthogonal group

We first introduce some notation. Let V be an 8-dimensional vector space over a field F of characteristic p ≥ 0, with basis e_{1}, …, e_{8} and Q the quadratic form on V defined by

Q : V → F, Q(∑_{i=1}^{8} x_{i}e_{i}) = ∑_{i=1}^{4} x_{i}x_{9−i} .

We denote by GO_{8}(F) the group of isometries of Q, the full orthogonal group, and by SO_{8}(F) the connected component of the identity in GO_{8}(F), of index 2. Thus SO_{8}(F) is a simple split algebraic group over F of type D_{4}. The subgroup of upper triangular matrices of GL_{8}(F) contains a Borel subgroup B of SO_{8}(F). More precisely, the unipotent radical of B is generated by the root subgroups

X_{i} := {x_{i}(t) | t ∈ F }, i = 1, …, 12,

where the x_{i}(t) are defined as in Table 1. Here E_{i,j}(t) denotes the matrix having 1’s on the diagonal and one further non-zero entry t in position (i, j).

Table 1. Root subgroups of SO_{8}(F)

\begin{matrix}x_{1}(t) = E_{1,2}(t) − E_{7,8}(t) & 1000 & x_{7}(t) = E_{2,5}(t) − E_{4,7}(t) & 0101 \\ x_{2}(t) = E_{2,3}(t) − E_{6,7}(t) & 0100 & x_{8}(t) = E_{1,4}(t) − E_{5,8}(t) & 1110 \\ x_{3}(t) = E_{3,4}(t) − E_{5,6}(t) & 0010 & x_{9}(t) = E_{2,6}(t) − E_{3,7}(t) & 0111 \\ x_{4}(t) = E_{3,5}(t) − E_{4,6}(t) & 0001 & x_{10}(t) = E_{1,5}(t) − E_{4,8}(t) & 1101 \\ x_{5}(t) = E_{1,3}(t) − E_{6,8}(t) & 1100 & x_{11}(t) = E_{1,6}(t) − E_{3,8}(t) & 1111 \\ x_{6}(t) = E_{2,4}(t) − E_{5,7}(t) & 0110 & x_{12}(t) = E_{1,7}(t) − E_{2,8}(t) & 1211 \end{matrix}

Explicit realization of G_{2}(q) 3

A maximal torus T in B is given by the set of diagonal matrices

T := {t = diag(t_{1}, t_{2}, t_{3}, t_{4}, t_{4}^{−1}, t_{3}^{−1}, t_{2}^{−1}, t_{1}^{−1}) | t_{i} ∈ F^{×}}.

The simple roots with respect to T are now α_{i}, i = 1, …, 4, with α_{i}(t) = t_{i}/t_{i+1} for i = 1, 2, 3 and α_{4}(t) = t_{3}t_{4}. In Table 1 we have also recorded the decomposition of the root corresponding to a root subgroup into the simple roots α_{1}, …, α_{4}. Note that the simple root α_{2} (with label 0100) is the one belonging to the central node in the Dynkin diagram of type D_{4}.

The group PSO_{8}(F) := SO_{8}(F)/Z(SO_{8}(F)) possesses an outer automorphism γ of order 3 induced by the graph automorphism of the Dynkin diagram D_{4} which cyclically permutes the nodes 1, 3 and 4 and fixes the middle node 2. The group PSO_{8}(F)^{γ} of fixed points in PSO_{8}(F) under γ is again a simple connected algebraic group over F, of type G_{2}. Note that γ does not stabilize the natural representation of SO_{8}(F). Nevertheless we can construct G_{2}(F) as a preimage G of PSO_{8}(F)^{γ} in SO_{8}(F).

The Borel subgroup B of SO_{8}(F) contains a Borel subgroup of G. Its unipotent radical is the product of the subgroups

X_{i,j,k} := {x_{i}(t)x_{j}(t)x_{k}(t) | t ∈ F }

where (i, j, k) ∈ {(1, 3, 4), (5, 6, 7), (8, 9, 10)}, together with the root subgroups X_{i} = {x_{i}(t) | t ∈ F } for i ∈ {2, 11, 12} (see for example Carter [3, Prop. 13.6.3]). A maximal torus of G consists of the elements

{t = diag(t_{1}, t_{2}, t_{1}t_{2}^{−1}, 1, 1, t_{1}^{−1}t_{2}, t_{2}^{−1}, t_{1}^{−1}) | t_{i} ∈ F^{×}}.

From this description we find that the simple roots for G_{2}(F) are now α, β, with α(t) := t_{1}/t_{2} and β(t) := t_{2}^{2}/t_{1}, and with corresponding root subgroups X_{α} := X_{1,3,4}, X_{β} := X_{2} respectively.

An easy calculation with the generators of root subgroups given above now shows that G leaves invariant the hyperplane V_{1} of V consisting of vectors with equal fourth and fifth coordinate, as well as the 1-dimensional subspace V_{2} of V spanned by e_{4} −e_{5}. Thus we obtain an induced action of G on V_{1}, respectively on V_{1}/V_{2} when char(F) = 2. This yields a faithful matrix representation Γ : G_{2}(F) ↪ GL_{n}(F) of G_{2}(F), of dimension n = 7 when char(F) ≠ 2, respectively of dimension n = 6 when char(F) = 2. It is well-known that the smallest possible degree of a faithful representation of G_{2}(F) is 7, respectively 6 if char(F) = 2, so our representation Γ is irreducible.

Remark 2.1. The matrices given in [4, p. 34] do not define a representation of G_{2}(2^{f}). Indeed, the matrix for h_{a}(t) does not have determinant 1, as it should have (since G_{2}(2^{f}) is simple for f > 1). Its second diagonal entry should be t^{−1}. Conjugating X_{a}(t) by h_{a}(t′) one sees that the middle off-diagonal entry of X_{a}(t) should be t^{2} instead of t. The commutator relations (see Carter [3, 12.4]; [4, (2.1)] contains misprints) then show that similarly in the matrices for X_{a+b}(t) and X_{2a+b}(t) the second non-zero off-diagonal entry t should be replaced by t^{2}. In this way one recovers the representation constructed above.

3. Finding a suitable element

Let first q = 2^{f} be even. Then an easy calculation shows that in our 6-dimensional representation Γ : G_{2}(F) → GL_{6}(F) constructed above, we have

x_{α}(t) = ( \begin{matrix}1 & t & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & t^{2} & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & t \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) , x_{β}(t) = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & t & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & t & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) ,

and the longest element of the Weyl group of G_{2}(F) is represented by

w_{0} = ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{matrix} ) .

We choose g := x_{α}(t)x_{β}(u)w_{0} ∈ G_{2}(F) and let

(1)  D := Γ(g) = ( \begin{matrix}0 & 0 & 0 & tu & t & 1 \\ 0 & 0 & 0 & u & 1 & 0 \\ 0 & t^{2}u & t^{2} & 1 & 0 & 0 \\ 0 & u & 1 & 0 & 0 & 0 \\ t & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{matrix} ) .

Proposition 3.1. Let q be even and D be defined as above. Then no proper subgroup of G_{2}(q) contains conjugates of all specializations of D.

Proof. We use the fact that all maximal subgroups of the finite groups G_{2}(q) are known by Cooperstein [4]. For q = 2 specializations into 𝔽_{8} yield elements of orders 7 and 12, and no maximal subgroup of G_{2}(2) contains elements of both orders. For q = 4 specializations into 𝔽_{4} yield elements of orders 13,15,21. The only maximal subgroup of order divisible by 7 · 13 is PSL_{2}(13), but its order is not divisible by 5, so we are done again.

Now let q ≥ 8. Let G be a subgroup of G_{2}(q) containing conjugates of all specializations of D. Let α ∈ 𝔽_{q^{2}} of order q + 1. Then the minimal polynomial of α over 𝔽_{q} has the form X^{2} + Tr(α)X + 1, where Tr(α) = α + α^{q} ∈ 𝔽_{q}. Thus any element of 𝔽_{q^{2}} of order q + 1 occurs as a root of a polynomial of the shape

X^{2} + vX + 1, v ∈ 𝔽_{q}.

Clearly, all elements of 𝔽_{q}^{×} also occur as zeros of such a polynomial. Now for v ∈ 𝔽_{q} consider the specialization

ψ_{v} : 𝔽_{q}[t, u] → 𝔽_{q}, t ↦ 0, u ↦ v.

Then the specialization ψ_{v}(D) of D has characteristic polynomial

X^{6} + (v^{2} + 1)X^{4} + (v^{2} + 1)X^{2} + 1 = (X + 1)^{2}(X^{2} + vX + 1)^{2}.

The 1-eigenspace of ψ_{v}(D) only has dimension 1 for v ≠ 0, so the order of ψ_{v}(D) is divisible by 2. By our above considerations, we hence find elements of orders 2(q+1) and 2(q − 1) as specializations of D. (This can also be seen as follows: if t = 0 then g specializes to

x_{β}(u)w_{0} = x_{β}(u)(w_{β}w_{α})^{3} = x_{β}(u)w_{β} · w′

where w′ = w_{α}w_{β}w_{α}w_{β}w_{α} has order 2, centralizes x_{β}(u)w_{β}, and x_{β}(u)w_{β} represents the element

( \begin{matrix}u & 1 \\ 1 & 0 \end{matrix} )

in the subgroup ⟨X_{β}, X_{−β}⟩ ≅ SL_{2}(q).)

Next, consider the specialization

ψ′_{v} : 𝔽_{q}[t, u] → 𝔽_{q}, t ↦ v, u ↦ 0.

Here, ψ′_{v}(D) has characteristic polynomial

(X^{2} + vX + 1)^{2}(X^{2} + v^{2}X + 1).

By the argument above, this again yields elements of orders 2(q − 1) and 2(q + 1). But note that this time these elements never have an eigenvalue 1, nor have any of their powers of order larger than 2. Thus G contains subgroups of order (q ± 1)^{2}. Theorem 2.3 in [4] shows that either G ≤ SL_{2}(q) × SL_{2}(q) or G = G_{2}(q).

Finally, consider the specialization

ψ′′_{v} : 𝔽_{q}[t, u] → 𝔽_{q}, t ↦ v, u ↦ 1.

The corresponding specialization of D has characteristic polynomial

(X^{3} + v^{2}X + 1)(X^{3} + v^{2}X^{2} + 1).

If X^{3} + v^{2}X + 1 is reducible over 𝔽_{q}, then it has a linear factor X + a, a ∈ 𝔽_{q}, and X^{3} + v^{2}X + 1 = (X + a)(X^{2} + aX + 1/a). Clearly, the case a = 0 is not possible, so for at least one of the q possibilities for v ∈ 𝔽_{q} the characteristic polynomial has an irreducible factor of degree 3. In this case, the specialization of D has order dividing q^{2} + q + 1, but not q − 1. Since SL_{2}(q) × SL_{2}(q) doesn’t contain such elements, we have G = G_{2}(q), as claimed. □

For odd q = p^{f} we again choose g := x_{α}(t)x_{β}(u)w_{0} ∈ G_{2}(F). With

x_{α}(t) = ( \begin{matrix}1 & t & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & t & −t^{2} & 0 & 0 \\ 0 & 0 & 0 & 1 & −2t & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & −t \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) , x_{β}(t) = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & t & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & −t & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) ,

and

w_{0} = ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ) ,

this gives

(2)  D := Γ(g) = ( \begin{matrix}0 & 0 & 0 & 0 & tu & −t & 1 \\ 0 & 0 & 0 & 0 & u & −1 & 0 \\ 0 & −t^{2}u & −t^{2} & −t & 1 & 0 & 0 \\ 0 & −2tu & −2t & −1 & 0 & 0 & 0 \\ 0 & u & 1 & 0 & 0 & 0 & 0 \\ −t & −1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} )

in this case. This matrix has separable characteristic polynomial

X^{7} + (t^{2} + 1) X^{6} − (2t^{2} + u^{2} + 3) X^{5} − (t^{4} + 3t^{2} + u^{2} + 3) X^{4}
+ (t^{4} + 3t^{2} + u^{2} + 3) X^{3} + (2t^{2} + u^{2} + 3) X^{2} − (t^{2} + 1) X − 1.

We need the following result:

Lemma 3.2. Let q > 3 be an odd prime power. Then there exists v ∈ 𝔽_{q} such that

X^{3} − (v^{2} + 2)X − 1

is irreducible over 𝔽_{q}.

Proof. Assume that f := X^{3} − (v^{2} + 2)X − 1 is reducible. Then f has a zero a ∈ 𝔽_{q}, and X^{3} − (v^{2} + 2)X − 1 = (X − a)(X^{2} + aX + a^{−1}). These zeros are just the first coordinates of the 𝔽_{q}-points on the elliptic curve E defined by U^{3} − (V^{2} + 2)U − 1. By the Weil bounds [8], E has at most q + 1 + 2\sqrt{q} points (u, v) over 𝔽_{q}. Clearly, with (u, v) the point (u, −v) also lies on E, hence there are at most q/2 + 1 + \sqrt{q} distinct values a which can occur as zeros of f.

Next, we estimate how often f splits completely into linear factors. This happens if in addition the discriminant (a^{3} − 4)/a of X^{2} + aX + a^{−1} is a square in 𝔽_{q}. Thus we need to count points on the 𝔽_{q}-curve C defined by the two equations

U^{3} − (V^{2} + 2)U − 1, U^{3} − W^{2}U − 4.

Subtracting these two equations we see that U lies in the function field 𝔽_{q}(V, W). Since both V, W have degree at most 2 over 𝔽_{q}(U), the curve C has genus at most 4. Moreover, the only singular point of C is the point with coordinates (4, 0, 0) in characteristic 5. Again by the Weil bounds [8] this means that C has at least q + 1 − 2 · 4\sqrt{q} − 6 points over 𝔽_{q}. For each such point, changing the sign of the V, W -coordinates again yields a point, hence there are at least (q − 5 − 8\sqrt{q})/4 distinct a ∈ 𝔽_{q} for which f splits completely. Thus we obtain at most

q/2 + 1 + \sqrt{q} − (q − 5 − 8\sqrt{q})/4 = (q + 9)/4 + 3\sqrt{q}

factorizations of f into a linear and a quadratic factor. The discriminant of f is a polynomial in v of degree 6, hence f is inseparable for at most six values of v. Apart from those, each completely splitting f accounts for three different values of a, so we obtain a total of at most

(q + 9)/4 + 3\sqrt{q} + ((q − 5 − 8\sqrt{q})/4 − 6)/3 + 6 = (2q + 35)/6 + 7/3\sqrt{q}

reducible polynomials when v runs over 𝔽_{q}. Hence there remain at least

(q + 1)/2 − ((2q + 35)/6 + 7/3\sqrt{q}) = (q − 32)/6 − 7/3\sqrt{q}

irreducible polynomials. This is positive for q ≥ 257. For the remaining prime powers 3 < q < 257 a computer check shows that the assertion is also satisfied. (For q = 5, 9 there is just one irreducible polynomial of the required shape, for q = 3 there is none.)

Note that the counting of singular points and of inseparable f was very rough and a more detailed analysis would have reduced the bound considerably. □

Proposition 3.3. Let q be odd and D be the matrix defined in (2). Then no proper subgroup of G_{2}(q) contains conjugates of all specializations of D.

Proof. Again all maximal subgroups of G_{2}(q) are known by work of Kleidman [5]. For q = 3 specializations into 𝔽_{9} yield elements of orders 7,9,13. The only maximal subgroup of G_{2}(3) of order divisible by 7 · 13 is PSL_{2}(13), but that has no elements of order 9. For q = 5, specialization into 𝔽_{5} yields element orders 7,20 and 31, thus we are done again.

For q ≥ 7 let G be a subgroup of G_{2}(q) containing conjugates of all specializations of D. We again consider the specialization

ψ_{v} : 𝔽_{q}[t, u] → 𝔽_{q}, t ↦ 0, u ↦ v.

Then the square of ψ_{v}(D) has characteristic polynomial

(X − 1)^{3}(X^{2} − (v^{2} + 2)X + 1)^{2}.

This gives rise to elements of orders q ± 1 in G. Similarly, the specialization

ψ′_{v} : 𝔽_{q}[t, u] → 𝔽_{q}, t ↦ v, u ↦ 0,

yields the characteristic polynomial

(X − 1)(X^{2} − (v^{4} + 4v^{2} + 2)X + 1)(X^{2} − (v^{2} + 2)X + 1)^{2}

for the image of D^{2}. So as in the previous proof we deduce that G must contain subgroups of orders (q ± 1)^{2}. Theorem A in [5] shows that either G is contained in the central product SL_{2}(q) ∘ SL_{2}(q), or G = G_{2}(q). Finally, for the specialization

ψ′′_{v} : 𝔽_{q}[t, u] → 𝔽_{q}, t ↦ v, u ↦ 1,

we obtain the characteristic polynomial

(X − 1)(X^{3} + (v^{2} + 2)X^{2} − 1)(X^{3} − (v^{2} + 2)X − 1)

for ϕ′′_{v}(D). Since q ≥ 7 is odd, Lemma 3.2 shows that there exists v ∈ 𝔽_{q} such that the degree 3 factors of this polynomial are irreducible over 𝔽_{q}. But SL_{2}(q) ∘ SL_{2}(q) does not contain such elements, hence we have G = G_{2}(q). □

4. The polynomials

It remains to determine generating polynomials for the G_{2}(q)-extensions whose existence is guaranteed by Theorem 1.1 in conjunction with Propositions 3.1 and 3.3.

Theorem 4.1. Let q = 2^{f} be a power of 2. Then the polynomial

X^{q^{6}} + u^{e_{2}}t^{e_{4}}X^{q^{5}} + (u^{e_{1}}t^{e_{1}} + u^{e_{3}}t^{e_{1}} + t^{e_{1}} + t^{e_{3}} + 1)X^{q^{4}}

+ u^{e_{2}}t^{e_{4}}(t^{q^{3}+q^{2}} + t^{q^{3}−q} + 1)X^{q^{3}} + t^{e_{1}}(u^{e_{1}}t^{q^{2}−1} + u^{e_{1}}t^{q^{2}+q} + u^{e_{1}} + u^{e_{3}} + 1)X^{q^{2}}

+ u^{e_{2}}t^{q^{4}+2q^{2}−q}X^{q} + u^{e_{1}}t^{q^{4}−1}X,

with e_{1} := q^{4} − q^{2}, e_{2} := q^{4} − q^{3}, e_{3} := q^{4} + q^{3}, e_{4} := q^{4} − q^{3} + 2q^{2}, has Galois group G_{2}(q) over 𝔽_{q}(t, u).

Proof. In Proposition 3.1 we have shown that the assumptions of Matzat’s theorem 1.1 are verified for the matrix D defined in (1). According to Matzat [6, §1], a generating polynomial for a field extension with group G_{2}(q) can now be obtained by solving the non-linear system of equations given by

y = D y^{q},

where y = (y_{1}, …, y_{6})^{t}, for one of the variables. Solving for y_{6} yields the equation displayed in the statement. □

By the Hilbert irreducibility theorem, there exist 1-parameter specializations of the polynomial in Theorem 4.1 with group G_{2}(q).

Example 4.2. By arguments similar to those used in the proof of Proposition 3.1 it can be checked that the polynomial

X^{64} + t^{24} X^{32}+(t^{36} + t^{12} + 1) X^{16} + (t^{30} + t^{36} + t^{24}) X^{8}
+(t^{24} + t^{36} + t^{27} + t^{30} + t^{12}) X^{4} + t^{30} X^{2} + t^{27} X

obtained by setting u = t has Galois group G_{2}(2) over 𝔽_{2}(t).

Theorem 4.3. Let q = p^{f} be an odd prime power. Then the polynomial

X^{q^{7}} + u^{e_{1}}t^{e_{4}}(t^{e_{6}} + 1)X^{q^{6}} − (t^{e_{2}}u^{e_{3}} + (t^{q^{5}+q^{2}} + t^{e_{2}})u^{e_{2}} + t^{e_{3}} + t^{e_{2}} + 1)X^{q^{5}}
− u^{e_{1}}t^{e_{4}}(t^{e_{5}}(u^{q^{4}+q^{3}} + u^{e_{5}}) + (t^{e_{6}} + 1)(t^{q^{4}+q^{3}} + t^{e_{5}} + 1))X^{q^{4}}
+ t^{e_{2}}(u^{e_{3}} + (t^{e_{6}} + 1)(t^{e_{6}} + t^{q^{3}−q} + 1)u^{e_{2}} + 1)X^{q^{3}}
+ u^{e_{1}}t^{q^{5}+q^{3}−2q^{2}}(u^{q^{4}+q^{3}} + (t^{q^{2}+q} + t^{q^{2}−1} + 1)u^{e_{5}} + t^{e_{6}} + 1)X^{q^{2}}
− u^{e_{2}}t^{q^{5}−q}(t^{e_{6}} + 1)X^{q} − u^{q^{5}−q^{2}}t^{q^{5}+q^{3}−q^{2}−1}X,

where e_{1} := q^{5} − q^{4}, e_{2} := q^{5} − q^{3}, e_{3} := q^{5} + q^{4}, e_{4} := q^{5} − q^{4} + q^{3} − q^{2}, e_{5} := q^{4} − q^{2}, e_{6} := q^{3} + q^{2}, has Galois group G_{2}(q) over 𝔽_{q}(t, u).

The proof is as for the preceding theorem, starting this time from the matrix D given in (2), solving for y_{7}, and using Proposition 3.3.

Remark 4.4. The sporadic simple Janko groups J_{1} and J_{2} are subgroups of G_{2}(11), respectively of G_{2}(4). It would be nice to find Galois extensions for these groups in characteristic 11 respectively 2 by the above method, possibly as specializations of the polynomials in Theorems 4.1 and 4.3.

Remark 4.5. The next smallest simple exceptional group is the one of type F_{4}. Its smallest faithful representation has dimension 26, respectively 25 in characteristic 3. In principle, the methods of this paper should make it possible to produce an F_{4}(q)- polynomial.

Remark 4.6. The group G_{2}(q), q odd, has q orbits on non-zero vectors in its 7dimensional representation. Thus, the polynomial f_{q}(t, u, X) in Theorem 4.3 has q factors, of degrees roughly q^{6}, and a linear factor. On the other hand, any specialization of f_{q} has factors of degree at most q^{2} + q + 1, the maximal element order in G_{2}(q). Thus, f_{q} seems a good candidate for testing factorization algorithms. Using Maple we have not been able to find the factorization of f_{q} for q = 3.

Similarly, for q even G_{2}(q) has a single orbit on the non-zero vectors of the 6dimensional module. Hence f_{q}(t, u, X) in Theorem 4.1 is irreducible apart from the trivial linear factor. Again Maple was not able to confirm this for q = 4.

References

[1] S. Abhyankar, N. Inglis, Galois groups of some vectorial polynomials, Trans. Amer. Math. Soc. 353 (2001), 2941–2869.

[2] S. Abhyankar, H. Popp, W. Seiler, Construction techniques for Galois coverings of the affine line. Proc. Indian Acad. Sci. 103 (1993), 103–126.

[3] R. Carter, Simple groups of Lie type, 2nd edition, Wiley, London, 1989.

[4] B. N. Cooperstein, Maximal subgroups of G_{2}(2^{n}), J. Algebra 70 (1981), 23–36.

[5] P. Kleidman, The maximal subgroups of the Chevalley groups G_{2}(q) with q odd, the Ree groups ^{2}G_{2}(q), and their automorphism groups, J. Algebra 117 (1988), 30–71.

[6] B. H. Matzat, Frobenius modules and Galois groups, preprint, Universität Heidelberg, July 2002.

[7] M. V. Nori, Unramified coverings of the affine line in positive characteristic, Algebraic geometry and its applications, Springer Verlag, New York, 1994, pp. 209–212.

[8] A. Weil, Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc. 55 (1949), 497–508.

Fachbereich Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany.

  E-mail address: malle@mathematik.uni-kassel.de