Galois groups of prime degree polynomials with nonreal roots

Details Galois groups of prime degree polynomials with nonreal roots Galois groups of prime degree polynomials with nonreal roots

2006-01-17

arxiv.org/abs/math/0601397v1

Lect. Notes in Computing, 13, 2005, 243--255

Mathematics

Group Theory

Scientific paper

In the process of computing the Galois group of a prime degree polynomial $f(x)$ over $\mathbb Q$ we suggest a preliminary checking for the existence of non-real roots. If $f(x)$ has non-real roots, then combining a 1871 result of Jordan and the classification of transitive groups of prime degree which follows from CFSG we get that the Galois group of $f(x)$ contains $A_p$ or is one of a short list. Let $f(x)\in \mathbb Q [x] $ be an irreducible polynomial of prime degree $p \geq 5$ and $r=2s$ be the number of non-real roots of $f(x)$. We show that if $s$ satisfies $s (s \log s + 2 \log s + 3) \leq p $ then $Gal (f)= A_p, S_p$.

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