structural:scholar_edition
Quadratic extensions

The degree two field extension Q (2) / Q {\displaystyle \mathbb {Q} ({\sqrt {2}})/\mathbb {Q} } has the Galois group Gal (Q (2) / Q) {\displaystyle \operatorname {Gal} (\mathbb {Q} ({\sqrt {2}})/\mathbb {Q})} with two elements, the identity automorphism and the automorphism σ {\displaystyle \sigma } which exchanges 2 {\displaystyle {\sqrt {2}}} and − 2 {\displaystyle -{\sqrt {2}}}. This example generalizes for a prime number p ∈ N. {\displaystyle p\in \mathbb {N}.}