A note on Galois groups of linearized polynomials. (English) Zbl 07813867
The goal of this paper is to extend the main result, Theorem 2, of R. Gow and G. McGuire [J. Number Theory 253, 368–377 (2023; Zbl 07734980)]. The main result of this paper is the following. Let \(q\) be a prime power and \(n\) be an odd prime. Let \(L(X)=\sum_{i=0}^na_iX^{q^i}\in {\mathbb F}_q[X]\), \(a_n\neq 0\), be a monic linearized polynomial over \({\mathbb F}_q\) of \(q\)-degree \(n\). Then the Galois group of \(L(X)/X-t\) over \({\mathbb F}_q(t)\) is \({\mathrm {GL}}_n(q)\) unless \(L(X)= X^{q^n}\).
The original result of Gow and McGuire is under the assumption that \(q\) is odd. The main result is a consequence of the following result. Let \(q\) be a prime power and \(n\) be a positive integer. Set \(L(X)=\sum_{i=m}^n a_iX^{q^i}\in {\mathbb F}_q[X]\), \(a_ma_n\neq 0\), where \(1\leq m\leq n-1\). Then \(q^m(q^m-1)(q^n-1)\) divides the order of the Galois group of \(L(X)/X-t\) over \({\mathbb F}_q (t)\).
The proof of the divisibility result uses Hensel’s Lemma.
The original result of Gow and McGuire is under the assumption that \(q\) is odd. The main result is a consequence of the following result. Let \(q\) be a prime power and \(n\) be a positive integer. Set \(L(X)=\sum_{i=m}^n a_iX^{q^i}\in {\mathbb F}_q[X]\), \(a_ma_n\neq 0\), where \(1\leq m\leq n-1\). Then \(q^m(q^m-1)(q^n-1)\) divides the order of the Galois group of \(L(X)/X-t\) over \({\mathbb F}_q (t)\).
The proof of the divisibility result uses Hensel’s Lemma.
Citations:
Zbl 07734980References:
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