8
$\begingroup$

On my homework I have been asked to compute the Galois group of a quintic. I have no idea how to do this, except

(a) I calculated that it was irreducible (brute-force)

(b) Since it is irreducible, its splitting field must have degree divisible by $5$

(c) The Galois group must be a subgroup of $S_5$.

There is also a fact that looks helpful, about $S_n$ only having one normal subgroup for $n \geq 5$. Does this mean there's only one possible Galois extension, or is that only if $S_n$ was already some sort of Galois group?

Aside from that, I have no idea how to do the problem, so can you help me?

Thanks!

P.S. The quintic in question is $x^5 + x - 1 \in \mathbb{Q}[x]$

  • 15
    $x^5 + x - 1 = (x^2-x+1) (x^3+x^2-1)$ is *reducible*.2011-05-13
  • 0
    Well, that is very good to know! Thanks.2011-05-13
  • 0
    Related: https://math.stackexchange.com/questions/1375747/2016-12-26

2 Answers 2