Kummer theorem

Let be the field of fractions of a Dedekind ring , let be an extension (cf. Extension of a field) of of degree , let be the integral closure of in , and let be a prime ideal in ; let , where and the elements constitute a basis for the -module ; finally, let be the irreducible polynomial of , let be the image of in the ring and let be the irreducible factorization of in . Then the prime ideal factorization of the ideal in is

with the degree of the polynomial equal to the degree of the extension of the field of residues.

Kummer's theorem makes it possible to determine the factorization of a prime ideal over an extension of the ground field in terms of the factorization in the residue class field of the irreducible polynomial of a suitable primitive element of the extension.

The theorem was first proved, for certain particular cases, by E.E. Kummer [1]; he used it to determine the factorization law in cyclotomic fields and in certain cyclic extensions of cyclotomic fields (cf. Cyclotomic field).

References

[1]	E.E. Kummer, "Zur Theorie der complexen Zahlen" J. Reine Angew. Math. , 35 (1847) pp. 319–326
[2]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)

References

[a1]	E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9