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
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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) |
Comments
References
[a1] | E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9 |
Kummer theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kummer_theorem&oldid=13665