Galois group of $x^4+px+p$

Asked Nov 25 '21 at 03:42

Active Nov 25 '21 at 03:42

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I am able to show the Galois group over $\mathbb Q$ of $x^4+px+p$ where $p$ is a prime can be $D_4$ or $C_4$ if and only if $p$ is $3$ or $5$, but how can I show that it is exactly $C_4$ when $p=5$ and is $D_4$ when $p=3$.

Is there any way without solving the equation?

asked Nov 25 '21 at 03:42

user782932

Do you have the explicit set of automorphisms? When $p = 5$, does one of the automorphisms generate all of them under (self-)composition? This should fail when $p = 3$ -- you should find one automorphism is an involution and the other(s) has(have) order $3$ ... – Eric Towers Nov 25 '21 at 03:45
@EricTowers I don't have the explicit automorphisms... I need to compute them I think – user782932 Nov 25 '21 at 03:50
Does this answer your question? "Standard" ways of telling if an irreducible quartic polynomial has Galois group C_4? – Parcly Taxel Nov 27 '21 at 13:16

Galois group of $x^4+px+p$

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