Namespaces
Variants
Actions

Difference between revisions of "Kummer theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (link)
(TeX done)
 
Line 1: Line 1:
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k0559901.png" /> be the [[field of fractions]] of a [[Dedekind ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k0559902.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k0559903.png" /> be an extension (cf. [[Extension of a field|Extension of a field]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k0559904.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k0559905.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k0559906.png" /> be the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k0559907.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k0559908.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k0559909.png" /> be a prime ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599010.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599012.png" /> and the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599013.png" /> constitute a basis for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599014.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599015.png" />; finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599016.png" /> be the irreducible polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599017.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599018.png" /> be the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599019.png" /> in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599020.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599021.png" /> be the irreducible factorization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599023.png" />. Then the prime ideal factorization of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599025.png" /> is
+
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
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599026.png" /></td> </tr></table>
+
\mathfrak{p}B = \mathfrak{P}_1^{e_1} \cdots \mathfrak{P}_r^{e_r}
 
+
$$
with the degree of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599027.png" /> equal to the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055990/k05599028.png" /> of the extension of the field of residues.
+
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.
 
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 [[#References|[1]]]; he used it to determine the factorization law in cyclotomic fields and in certain cyclic extensions of cyclotomic fields (cf. [[Cyclotomic field|Cyclotomic field]]).
+
The theorem was first proved, for certain particular cases, by E.E. Kummer [[#References|[1]]]; he used it to determine the factorization law in cyclotomic fields and in certain cyclic extensions of [[cyclotomic field]]s.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.E. Kummer,  "Zur Theorie der complexen Zahlen"  ''J. Reine Angew. Math.'' , '''35'''  (1847)  pp. 319–326</TD></TR><TR><TD valign="top">[2]</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">  E.E. Kummer,  "Zur Theorie der complexen Zahlen"  ''J. Reine Angew. Math.'' , '''35'''  (1847)  pp. 319–326 {{DOI|10.1515/crll.1847.35.319}}</TD></TR>
 
+
<TR><TD valign="top">[2]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1986) {{ZBL|0645.12001}}</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
Line 18: Line 19:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Weiss,  "Algebraic number theory" , McGraw-Hill  (1963)  pp. Sects. 4–9</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Weiss,  "Algebraic number theory" , McGraw-Hill  (1963)  pp. Sects. 4–9 {{ZBL|0115.03601}}</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 11:31, 17 September 2017

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.

References

[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

Comments

References

[a1] E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9 Zbl 0115.03601
How to Cite This Entry:
Kummer theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kummer_theorem&oldid=41875
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