| 词条 | Bauerian extension |
| 释义 |
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.[1] 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.[2] See also
References1. ^Narkiewicz (1990) p.416 2. ^Narkiewicz (1990) p.394
1 : Theorems in algebraic number theory |
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