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 $\mathrm{l}\mathrm{c}\mathrm{m}(n_1,\dots ,n_r)$.

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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.