Kummer theorem

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Let $k$ be the field of fractions of a Dedekind ring $A$, let $K$ be a field extension of $k$ of degree $n$, let $B$ be the integral closure of $A$ in $K$, and let $\mathfrak{p}$ be a prime ideal in $A$; suppose that $K = k[\theta]$, where $\theta \in B$ and the elements $1,\theta,\ldots,\theta^{n-1}$ constitute a basis for the $A$-module $B$; finally, let $f(x)$ be the irreducible polynomial of $\theta$ over $k$, let $f^*(x)$ be the image of $f(x)$ in the ring $A/\mathfrak{p}[x]$ and let $f^*(x) = f_1^*(x)^{e_1}\cdots f_r^*(x)^{e_r}$ be the irreducible factorization of $f^*(x)$ in $A/\mathfrak{p}[x]$. Then the prime ideal factorization of the ideal $\mathfrak{p}B$ in $B$ is $$ \mathfrak{p}B = \mathfrak{P}_1^{e_1} \cdots \mathfrak{P}_r^{e_r} $$ with the degree of the polynomial $f_i^*(x)$ equal to the degree $[B/\mathfrak{P}_i : A/\mathfrak{p}]$ of the extension of the residue fields.

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.


[1] E.E. Kummer, "Zur Theorie der complexen Zahlen" J. Reine Angew. Math. , 35 (1847) pp. 319–326 DOI 10.1515/crll.1847.35.319
[2] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986) Zbl 0645.12001



[a1] E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9 Zbl 0115.03601
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Kummer theorem. Encyclopedia of Mathematics. URL:
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