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(L/K) to be the set of primes p of K which have a factor P with inertial degree one (that is, the residue field of P has the same order as the residue field of p). Bauer's theorem states that if M/K is a finite degree Galois extension, then P(M/K) ⊇ P(L/K) 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(F/K) ⊇ P(L/K) 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.
See also Splitting of prime ideals in Galois extensions
References
Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. Vol. 62 (2nd printing of 1st ed.). Springer-Verlag. p. 86. ISBN 3-540-63003-1. Zbl 0819.11044. Narkiewicz, Władysław (1990). Elementary and analytic theory of numbers (Second, substantially revised and extended ed.). Springer-Verlag. ISBN 3-540-51250-0. Zbl 0717.11045.