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An [[Extension of a field|extension of a field]] $k$ of characteristic $p \ge 0$, of the type
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An [[extension of a field]] $k$ of characteristic $p \ge 0$, of the type
$$
+
\begin{equation}\label{eq:1}
 
K = k(a_1^{1/n},\ldots,a_t^{1/n})
 
K = k(a_1^{1/n},\ldots,a_t^{1/n})
$$
+
\end{equation}
 
where $a_1,\ldots,a_t \in k$, $n$ is some natural number, and it is assumed that $k$ contains a primitive $n$-th root of unity $\zeta_n$ (in particular, if $p \ne 0$ then $n$ is prime to $p$). Kummer extensions are named after E. Kummer, who first studied extensions of the type $\mathbf{Q}(\zeta_n,a^{1/n})$, where $\mathbf{Q}$ is the field of rational numbers and $a \in \mathbf{Q}$.
 
where $a_1,\ldots,a_t \in k$, $n$ is some natural number, and it is assumed that $k$ contains a primitive $n$-th root of unity $\zeta_n$ (in particular, if $p \ne 0$ then $n$ is prime to $p$). Kummer extensions are named after E. Kummer, who first studied extensions of the type $\mathbf{Q}(\zeta_n,a^{1/n})$, where $\mathbf{Q}$ is the field of rational numbers and $a \in \mathbf{Q}$.
  
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$$
 
$$
 
where $\mu(n)$ is the subgroup of $k^*$ generated by $\zeta_n$. If $\sigma \in \mathrm{Gal}(K/k)$ and $a \in A(K/k$, the pairing is defined by the formula $(\sigma,a) = (a^{1/n})^{\sigma-1}$, where $a \in k$, and $a^{1/n} \in B$ is some representative of the element $a$. The pairing defines a canonical isomorphism.
 
where $\mu(n)$ is the subgroup of $k^*$ generated by $\zeta_n$. If $\sigma \in \mathrm{Gal}(K/k)$ and $a \in A(K/k$, the pairing is defined by the formula $(\sigma,a) = (a^{1/n})^{\sigma-1}$, where $a \in k$, and $a^{1/n} \in B$ is some representative of the element $a$. The pairing defines a canonical isomorphism.
 +
\begin{equation}\label{eq:2}
 +
\mathrm{Gal}(K/k) \stackrel{\phi}{\cong} \mathrm{Hom}(A(K/k),\mu(n)
 +
\end{equation}
 +
 +
In other words, any automorphism $\sigma \in \mathrm{Gal}(K/k)$ is defined by its action on the roots $a_i^{1/n}$ in \eqref{eq:1}, and this action may be arbitrary, provided that the roots $a_i^{1/n}$ are independent. In particular, if $\mathrm{Gal}(K/k)$ is a cyclic group, then $K = k(a^{1/n})$, where $a \in k^*$.
 +
 +
Let $k$ be a [[normal extension]] of a field $k_0$ and let $K/k$ be a Kummer extension. Then the field $K$ is normal over $k_0$ if and only if $A(K/k$ is mapped into itself by $\mathrm{Gal}(k/k_0)$. In that case the isomorphism \eqref{eq:2} is a $\mathrm{Gal}(k/k_0)$-isomorphism, i.e. if $\tau \in \mathrm{Gal}(k/k_0)$, $\sigma \in \mathrm{Gal}(K/k)$ and if
 
$$
 
$$
\mathrm{Gal}(K/k) \cong \mathrm{Hom}(A(K/k),\mu(n)
+
\phi(\sigma) = \chi : A(K/k) \rightarrow \mu(n)
 
$$
 
$$
 +
then $\phi(\sigma^\tau) = \tau\chi$, where $\tau\chi(a) = \tau(\chi(\tau^{-1}(a)))$. (The group $\mathrm{Gal}(k/k_0)$ acts on $\mathrm{Gal}(K/k)$ via conjugation in $\mathrm{Gal}(K/k_0)$.) By the above proposition, many problems concerning Abelian extensions of exponent $n$ of a field $k$ can be reduced to the theory of Kummer extensions even if $\zeta_n \not\in k$. To be precise: If $K/k$ is such an extension, then $K(\zeta_n)/k(\zeta_n)$ is a Kummer extension, and its Kummer group is characterized by the condition: If $\tau \in \mathrm{Gal}(k(\zeta_n)/k)$ and $a \in A(K(\zeta_n)/k(\zeta_n))$, then $\tau(a) = a^i$, where $i$ is a natural number which is defined modulo $n$ by the condition $\tau(\zeta_n) = \zeta_n^i$.
  
In other words, any automorphism $\sigma \in \mathrm{Gal}(K/k)$ is defined by its action on the roots $a_i^{1/n}$ in (1), and this action may be arbitrary, provided that the roots $a_i^{1/n}$ are independent. In particular, if $\mathrm{Gal}(K/k)$ is a cyclic group, then $K = k(a^{1/n})$, where $a \in k^*$.
+
The main results concerning Kummer extensions may be derived as corollaries of the [[Hilbert theorem]] on cyclic extensions, according to which the one-dimensional [[Galois cohomology]] group $H^1(\mathrm{Gal}(K/k), K^*)$ is trivial.
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596044.png" /> be a [[Normal extension|normal extension]] of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596045.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596046.png" /> be a Kummer extension. Then the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596047.png" /> is normal over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596048.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596049.png" /> is mapped into itself by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596050.png" />. In that case the isomorphism (2) is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596051.png" />-isomorphism, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596053.png" /> and if
 
 
 
<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/k055960/k05596054.png" /></td> </tr></table>
 
 
 
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596056.png" />. (The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596057.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596058.png" /> via conjugation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596059.png" />.) By the above proposition, many problems concerning Abelian extensions of exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596060.png" /> of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596061.png" /> can be reduced to the theory of Kummer extensions even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596062.png" />. To be precise: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596063.png" /> is such an extension, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596064.png" /> is a Kummer extension, and its Kummer group is characterized by the condition: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596068.png" /> is a natural number which is defined modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596069.png" /> by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596070.png" />.
 
 
 
The main results concerning Kummer extensions may be derived as corollaries of the [[Hilbert theorem|Hilbert theorem]] on cyclic extensions, according to which the one-dimensional [[Galois cohomology|Galois cohomology]] group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596071.png" /> is trivial.
 
  
The theory of Kummer extensions carries over to the case of infinite Abelian extensions of exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596072.png" />. When this is done, the Kummer pairing establishes a [[Pontryagin duality|Pontryagin duality]] between the profinite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596073.png" /> (endowed with the [[Krull topology]]) and the discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596074.png" /> (see [[#References|[1]]], [[#References|[2]]]).
+
The theory of Kummer extensions carries over to the case of infinite Abelian extensions of exponent $n$. When this is done, the Kummer pairing establishes a [[Pontryagin duality]] between the profinite group $\mathrm{Gal}(K/k)$ (endowed with the [[Krull topology]]) and the discrete group $A(K/k)$ (see [[#References|[1]]], [[#References|[2]]]).
  
The theory of Kummer extensions, also known as Kummer theory, has an analogue in the case of extensions (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596075.png" /> (Artin–Schreier theory). The role of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596076.png" /> in that situation is played by the additive group of the prime subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596077.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596078.png" />. The main result of the theory is: Any Abelian extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596079.png" /> of exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596080.png" /> of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596081.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596082.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596083.png" /> are roots of equations of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596084.png" /> (see [[#References|[1]]]). There is also a generalization of this theory, due to E. Witt, to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596085.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596086.png" />, using Witt vectors (cf. [[Witt vector|Witt vector]]).
+
The theory of Kummer extensions, also known as Kummer theory, has an analogue in the case of extensions \eqref{eq:1} with $n=p$ (cf. the [[Artin–Schreier theorem]]). The role of the group $\mu(n)$ in that situation is played by the additive group of the prime subfield $\mathbf{F}_p$ of $k$. The main result of the theory is: Any Abelian extension $K$ of exponent $p$ of a field $k$ is of the form $k(\beta_1,\ldots,\beta_t)$, where $\beta_1,\ldots,\beta_t$ are roots of equations of the type $x^p -x = a$ (see [[#References|[1]]]). There is also a generalization of this theory, due to E. Witt, to the case $n=p^s$, where $s>1$, using [[Witt vector]]s.
  
Finally, an attempt has been made to construct a non-Abelian "Kummer theory" [[#References|[3]]], where the multiplicative group of the field is replaced by the matrix group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055960/k05596087.png" />.
+
Finally, an attempt has been made to construct a non-Abelian "Kummer theory" [[#References|[3]]], where the multiplicative group of the field is replaced by the matrix group $\mathrm{GL}(n,k)$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1968) {{MR|0911121}} {{MR|0255512}} {{MR|0215665}} {{ZBL|0645.12001}} {{ZBL|0153.07403}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Takahashi, "Generation of Galois extensions by matrix roots" ''J. Math. Soc. Japan'' , '''20''' : 1–2 (1968) pp. 365–370 {{MR|0224596}} {{ZBL|0182.07503}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1968) {{MR|0911121}} {{MR|0255512}} {{MR|0215665}} {{ZBL|0645.12001}} {{ZBL|0153.07403}} </TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top"> S. Takahashi, "Generation of Galois extensions by matrix roots" ''J. Math. Soc. Japan'' , '''20''' : 1–2 (1968) pp. 365–370 {{MR|0224596}} {{ZBL|0182.07503}} </TD></TR>
 +
</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. 4, §4 {{MR|0819231}} {{ZBL|0587.12001}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. 4, §4 {{MR|0819231}} {{ZBL|0587.12001}} </TD></TR>
 +
</table>
  
{{TEX|part}}
+
{{TEX|done}}

Latest revision as of 20:00, 18 September 2017

An extension of a field $k$ of characteristic $p \ge 0$, of the type \begin{equation}\label{eq:1} K = k(a_1^{1/n},\ldots,a_t^{1/n}) \end{equation} where $a_1,\ldots,a_t \in k$, $n$ is some natural number, and it is assumed that $k$ contains a primitive $n$-th root of unity $\zeta_n$ (in particular, if $p \ne 0$ then $n$ is prime to $p$). Kummer extensions are named after E. Kummer, who first studied extensions of the type $\mathbf{Q}(\zeta_n,a^{1/n})$, where $\mathbf{Q}$ is the field of rational numbers and $a \in \mathbf{Q}$.

The main result of the theory of Kummer extensions is that if the field $k$ contains a primitive root $\zeta_n$, then a finite extension $K/k$ is Kummer (for a given $n$) if and only if $K/k$ is a normal Abelian extension and the Galois group $\mathrm{Gal}(K/k)$ is annihilated by $n$. Any Kummer extension of a field $k$ is completely determined by its Kummer group $A(K/k) = B/k^*$, where $k^*$ is the multiplicative group of $k$ and $$ B = \{ x \in K^* : x^n \in k^* \} \ . $$

There exists a non-degenerate Kummer pairing, i.e. a mapping $$ \mathrm{Gal}(K/k) \times A(K/k) \rightarrow \mu(n) $$ where $\mu(n)$ is the subgroup of $k^*$ generated by $\zeta_n$. If $\sigma \in \mathrm{Gal}(K/k)$ and $a \in A(K/k$, the pairing is defined by the formula $(\sigma,a) = (a^{1/n})^{\sigma-1}$, where $a \in k$, and $a^{1/n} \in B$ is some representative of the element $a$. The pairing defines a canonical isomorphism. \begin{equation}\label{eq:2} \mathrm{Gal}(K/k) \stackrel{\phi}{\cong} \mathrm{Hom}(A(K/k),\mu(n) \end{equation}

In other words, any automorphism $\sigma \in \mathrm{Gal}(K/k)$ is defined by its action on the roots $a_i^{1/n}$ in \eqref{eq:1}, and this action may be arbitrary, provided that the roots $a_i^{1/n}$ are independent. In particular, if $\mathrm{Gal}(K/k)$ is a cyclic group, then $K = k(a^{1/n})$, where $a \in k^*$.

Let $k$ be a normal extension of a field $k_0$ and let $K/k$ be a Kummer extension. Then the field $K$ is normal over $k_0$ if and only if $A(K/k$ is mapped into itself by $\mathrm{Gal}(k/k_0)$. In that case the isomorphism \eqref{eq:2} is a $\mathrm{Gal}(k/k_0)$-isomorphism, i.e. if $\tau \in \mathrm{Gal}(k/k_0)$, $\sigma \in \mathrm{Gal}(K/k)$ and if $$ \phi(\sigma) = \chi : A(K/k) \rightarrow \mu(n) $$ then $\phi(\sigma^\tau) = \tau\chi$, where $\tau\chi(a) = \tau(\chi(\tau^{-1}(a)))$. (The group $\mathrm{Gal}(k/k_0)$ acts on $\mathrm{Gal}(K/k)$ via conjugation in $\mathrm{Gal}(K/k_0)$.) By the above proposition, many problems concerning Abelian extensions of exponent $n$ of a field $k$ can be reduced to the theory of Kummer extensions even if $\zeta_n \not\in k$. To be precise: If $K/k$ is such an extension, then $K(\zeta_n)/k(\zeta_n)$ is a Kummer extension, and its Kummer group is characterized by the condition: If $\tau \in \mathrm{Gal}(k(\zeta_n)/k)$ and $a \in A(K(\zeta_n)/k(\zeta_n))$, then $\tau(a) = a^i$, where $i$ is a natural number which is defined modulo $n$ by the condition $\tau(\zeta_n) = \zeta_n^i$.

The main results concerning Kummer extensions may be derived as corollaries of the Hilbert theorem on cyclic extensions, according to which the one-dimensional Galois cohomology group $H^1(\mathrm{Gal}(K/k), K^*)$ is trivial.

The theory of Kummer extensions carries over to the case of infinite Abelian extensions of exponent $n$. When this is done, the Kummer pairing establishes a Pontryagin duality between the profinite group $\mathrm{Gal}(K/k)$ (endowed with the Krull topology) and the discrete group $A(K/k)$ (see [1], [2]).

The theory of Kummer extensions, also known as Kummer theory, has an analogue in the case of extensions \eqref{eq:1} with $n=p$ (cf. the Artin–Schreier theorem). The role of the group $\mu(n)$ in that situation is played by the additive group of the prime subfield $\mathbf{F}_p$ of $k$. The main result of the theory is: Any Abelian extension $K$ of exponent $p$ of a field $k$ is of the form $k(\beta_1,\ldots,\beta_t)$, where $\beta_1,\ldots,\beta_t$ are roots of equations of the type $x^p -x = a$ (see [1]). There is also a generalization of this theory, due to E. Witt, to the case $n=p^s$, where $s>1$, using Witt vectors.

Finally, an attempt has been made to construct a non-Abelian "Kummer theory" [3], where the multiplicative group of the field is replaced by the matrix group $\mathrm{GL}(n,k)$.

References

[1] S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[2] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1968) MR0911121 MR0255512 MR0215665 Zbl 0645.12001 Zbl 0153.07403
[3] S. Takahashi, "Generation of Galois extensions by matrix roots" J. Math. Soc. Japan , 20 : 1–2 (1968) pp. 365–370 MR0224596 Zbl 0182.07503


Comments

The theory of Kummer extensions fits, of course, in the general framework of class field theory, cf. [a1] for details.

References

[a1] J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. 4, §4 MR0819231 Zbl 0587.12001
How to Cite This Entry:
Kummer extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kummer_extension&oldid=41890
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