Inertial prime number

inert prime number, in an extension $ K / \mathbf Q $

A prime number $ p $ such that the principal ideal generated by $ p $ remains prime in $ K / \mathbf Q $, where $ K $ is a finite extension of the field of rational numbers $ \mathbf Q $; in other words, the ideal $ ( p) $ is prime in $ B $, where $ B $ is the ring of integers of $ K $. In this case one also says that $ p $ is inert in the extension $ K / \mathbf Q $. By analogy, a prime ideal $ \mathfrak p $ of a Dedekind ring $ A $ is said to be inert in the extension $ K / k $, where $ k $ is the field of fractions of $ A $ and $ K $ is a finite extension of $ k $, if the ideal $ \mathfrak p B $, where $ B $ is the integral closure of $ A $ in $ K $, is prime.

If $ K / k $ is a Galois extension with Galois group $ G $, then for any ideal $ \mathfrak P $ of the ring $ B \subset K $, a subgroup $ T _ {\mathfrak P } $ of the decomposition group $ G _ {\mathfrak P } $ of the ideal $ \mathfrak P $ is defined which is called the inertia group (see Ramified prime ideal). The extension $ K ^ {T _ {\mathfrak P } } / K ^ {G _ {\mathfrak P } } $ is a maximal intermediate extension in $ K / k $ in which the ideal $ \mathfrak P \cap K ^ {G _ {\mathfrak P } } $ is inert.

In cyclic extensions of algebraic number fields there always exist infinitely many inert prime ideals.

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Let $ K / k $ be a Galois extension with Galois group $ G $. Let $ \mathfrak P $ be a prime ideal of (the ring of integers $ A _ {K} $) of $ K $. The decomposition group of $ \mathfrak P $ is defined by $ G _ {\mathfrak P } = \{ {\sigma \in G } : {\mathfrak P ^ \sigma = \mathfrak P } \} $. The subgroup $ I _ {\mathfrak P } = \{ {\sigma \in G _ {\mathfrak P } } : {a ^ \sigma \equiv a \mathop{\rm mod} \mathfrak P \textrm{ for all } a \in B } \} $ is the inertia group of $ \mathfrak P $ over $ k $. It is a normal subgroup of $ G _ {\mathfrak P } $. The subfields of $ K $ which, according to Galois theory, correspond to $ G _ {\mathfrak P } $ and $ I _ {\mathfrak P } $, are called respectively the decomposition field and inertia field of $ \mathfrak P $.