1. On Galois groups of linearized polynomials related to the special linear group of prime degree.
- Author
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Gow, Rod and McGuire, Gary
- Subjects
- *
POLYNOMIAL rings , *FINITE fields , *POLYNOMIALS , *VECTOR spaces - Abstract
Let F be any field of prime characteristic p and let q be a power of p. We assume that F contains the finite field of order q. A q -polynomial L over F is an element of the polynomial ring F [ x ] with the property that all exponents are a power of q. We assume that the coefficient of the x term of L is nonzero. We investigate the Galois group G of L over F , under the assumption that L (x) / x is irreducible in F [ x ]. It is well known that if L has q -degree n , its roots are an n -dimensional vector space over the field of order q and G acts linearly on this space. Our main theorem is the following. We consider a monic q -polynomial L whose q -degree is a prime, r , say, with L (x) / x irreducible over F. Assuming that the coefficient of the x term of L is (− 1) r , we show that the Galois group of L over F must be the special linear group S L (r , q) of degree r over the field of order q when q > 2. We also prove a projective version for the projective special linear group P S L (r , q) , and we present the analysis when q = 2. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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