Difference between revisions of "Inertial prime number"
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| − | + | ''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|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. | In cyclic extensions of algebraic number fields there always exist infinitely many inert prime ideals. | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lang, "Algebraic number theory" , Addison-Wesley (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1986)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lang, "Algebraic number theory" , Addison-Wesley (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1986)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | Let | + | 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 $. | ||
Latest revision as of 22:12, 5 June 2020
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.
References
| [1] | S. Lang, "Algebraic number theory" , Addison-Wesley (1970) |
| [2] | H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959) |
| [3] | J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986) |
Comments
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 $.
How to Cite This Entry:
Inertial prime number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inertial_prime_number&oldid=13151
Inertial prime number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inertial_prime_number&oldid=13151
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article