Editing Galois group (section)

=== Computational tools ===

==== Cardinality of the Galois group and the degree of the field extension ====
One of the basic propositions required for completely determining the Galois group<ref name=":0">{{Cite web| url=http://abstract.ups.edu/download/aata-20100827.pdf |archive-url=https://web.archive.org/web/20111218100650/http://abstract.ups.edu/download/aata-20100827.pdf |archive-date=2011-12-18 |url-status=live |title=Abstract Algebra| pages=372–377}}</ref> of a finite field extension is the following: Given a polynomial <math>f(x) \in F[x]</math>, let <math>E/F</math> be its splitting field extension. Then the order of the Galois group is equal to the degree of the field extension; that is,

:<math>\left|\operatorname{Gal}(E/F)\right| = [E:F]</math>

==== Eisenstein's criterion ====
A useful tool for determining the Galois group of a polynomial comes from [[Eisenstein's criterion]]. If a polynomial <math>f \in F[x]</math> factors into irreducible polynomials <math>f = f_1\cdots f_k</math> the Galois group of <math>f</math> can be determined using the Galois groups of each <math>f_i</math> since the Galois group of <math>f</math> contains each of the Galois groups of the <math>f_i.</math>