Bauerian extension
In mathematics, in the field of algebraic number theory, a Bauerian extension is a field extension of an algebraic number field which is characterized by the prime ideals with inertial degree one in the extension.
For a finite degree extension L/''K of an algebraic number field K'' we define P to be the set of primes p of K which have a factor P with inertial degree one.
Bauer's theorem states that if M/''K is a finite degree Galois extension, then P'' ⊇ P if and only if M ⊆ L. In particular, finite degree Galois extensions N of K are characterised by set of prime ideals which split completely in N.
An extension F/''K is Bauerian if it obeys Bauer's theorem: that is, for every finite extension L'' of K, we have P ⊇ P if and only if L contains a subfield K-isomorphic to F.
All field extensions of degree at most 4 over Q are Bauerian.
An example of a non-Bauerian extension is the Galois extension of Q by the roots of 2x5 − 32x + 1, which has Galois group S5.