Inertial prime number

inert prime number, in an extension

A prime number such that the principal ideal generated by remains prime in , where is a finite extension of the field of rational numbers ; in other words, the ideal is prime in , where is the ring of integers of . In this case one also says that is inert in the extension . By analogy, a prime ideal of a Dedekind ring is said to be inert in the extension , where is the field of fractions of and is a finite extension of , if the ideal , where is the integral closure of in , is prime.

If is a Galois extension with Galois group , then for any ideal of the ring , a subgroup of the decomposition group of the ideal is defined which is called the inertia group (see Ramified prime ideal). The extension is a maximal intermediate extension in in which the ideal 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 be a Galois extension with Galois group . Let be a prime ideal of (the ring of integers ) of . The decomposition group of is defined by . The subgroup is the inertia group of over . It is a normal subgroup of . The subfields of which, according to Galois theory, correspond to and , are called respectively the decomposition field and inertia field of .