Galois group of a polynomial

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This term is related to: Galois theory
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Definition

For $K$ a field, let $f(X)\in K[X]$ be a separable polynomial with coefficients in $K$, and $L$ a splitting field for $f$ over $K$. Then the Galois group (in the sense of field extensions) of $L$ over $K$ acting on the roots $\alpha _{1},\dots ,\alpha _{n}$ of $f$ determines an injective group homomorphism $\iota :G\to S_{n}$. (Here, $S_{n}$ is the symmetric group on $n$ letters).

Then the image of $\iota$ is a subgroup of $S_{n}$, and is termed the Galois group of the polynomial $f(X)$.

Examples

• The polynomial $f(X)=(X^{2}-2)(X^{2}-3)\in \mathbb {Q} [X]$ has splitting field $\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})$ over $\mathbb {Q}$. It can be shown that this field extension has Galois group isomorphic to the Klein four-group.

• The polynomial $X^{n}-X-1\in \mathbb {Q} [X]$ has Galois group the symmetric group on $n$ letters.

• Let $p$ be a prime, and $\mathbb {F} _{p}$ denote the finite field with $p$ elements. Let $f\in \mathbb {F} _{p}[X]$ be a separable polynomial that factors into irreducible factors of degrees $n_{1},\dots n_{r}$, and denote the degree of $f$ as $n:=\sum _{i}n_{i}$. Then the Galois group of $f$ over $\mathbb {F} _{p}$ viewed as a subgroup of the symmetric group $S_{n}$ is generated by a single permutation with cycle type $(n_{1},\dots ,n_{r})$. Thus, it is cyclic of order $\mathrm {lcm} (n_{1},\dots ,n_{r})$.

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Applications and results

Insolvability of the quintic

It can be shown that an irreducible polynomial $f$ is solvable in terms of radicals if and only if its Galois group is a solvable group. But there exist degree five polynomials with Galois group alternating group:A5 and symmetric group:S5, which are not solvable groups. Thus in general, the quintic is not solvable in radicals.