View source for Kummer extension

An [[Extension of a field|extension of a field]] $k$ of characteristic $p \ge 0$, of the type
$$
K = k(a_1^{1/n},\ldots,a_t^{1/n})
$$
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.
$$
\mathrm{Gal}(K/k) \cong \mathrm{Hom}(A(K/k),\mu(n)
$$

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 $a \in k^*$.

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, 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]]).

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" />.

====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>



====Comments====
The theory of Kummer extensions fits, of course, in the general framework of [[Class field theory|class field theory]], cf. [[#References|[a1]]] for details.

====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>

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