So here is my question. If a polynomial $f(x)\in\mathbb{Q}[x]$ is separable of degree $n$ and has Galois group $G\subseteq S_n$.Is it true that if $\bar{f}\in\mathbb{Z}_p[x]$ is separable then $\bar{G}\simeq G\subseteq S_n$. It seems true, but I am not sure. Any help is appreciated.
Galois groups of a a seperable polynomial in a splitting field over $\mathbb{Q}$ and $\mathbb{Z}_p$
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abstract-algebra
field-theory
galois-theory
finite-fields
1 Answers
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No. For instance, $f(x)=x^2+1$ has Galois group $\mathbb{Z}/2\mathbb{Z}$ over $\mathbb{Q}$, but $\bar{f}(x)=(x+2)(x+3)$ in $\mathbb{F}_5[x]$, hence its Galois group over $\mathbb{F}_5$ is trivial.
Moreover, since the Galois group of any finite extension of a finite field is cyclic, you don't have much hope in general that the Galois groups of $f$ and $\bar{f}$ will be isomorphic.
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0Thanks. If its not too much of a bother, what if we weaken the condition to $\bar{G}\leq G$. Then would the above assertion be true? – 2017-02-09
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1Yes. Milne's Field Theory notes (which are freely available on his website) has a nice discussion of this, and how it can be used to try to determine $G$. – 2017-02-09