Galois extensions for cyclic group:Z2

Given a Galois extension of fields, the Galois group for the extension is the cyclic group of order two if and only if the field extension is a quadratic extension.

Over a finite field

Over a finite field, there is exactly one quadratic extension.

Case where the field does not have characteristic two

In this case, the quadratic extension can be obtained by adjoining a polynomial of the form $x^2-a$ where $a$ is a nonzero element of the field that is not a square. For prime fields, such an element is termed a quadratic nonresidue.

Case where the field has characteristic two

This case is a little trickier, since no polynomial of the form $x^2-a$ is irreducible. Rather, an irreducible polynomial is of the form $x^2+x+a$.