Difference between revisions of "Kummer extension"
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− | An [[Extension of a field|extension of a field]] | + | 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 | |
− | + | $$ | |
− | + | B = \{ x \in K^* : x^n \in k^* \} \ . | |
− | + | $$ | |
− | The main result of the theory of Kummer extensions is that if the field | ||
− | |||
− | |||
There exists a non-degenerate Kummer pairing, i.e. a mapping | 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. | ||
+ | $$ | ||
+ | \mathrm{Gal}(K/k) \cong \mathrm{Hom}(B,\mu(n) | ||
+ | $$ | ||
− | + | 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 $a \in k^*$. | |
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− | |||
− | |||
− | |||
− | |||
− | In other words, any automorphism | ||
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 | 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 | ||
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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}} |
Revision as of 16:30, 18 September 2017
An 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. $$ \mathrm{Gal}(K/k) \cong \mathrm{Hom}(B,\mu(n) $$
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 be a normal extension of a field
and let
be a Kummer extension. Then the field
is normal over
if and only if
is mapped into itself by
. In that case the isomorphism (2) is a
-isomorphism, i.e. if
,
and if
![]() |
then , where
. (The group
acts on
via conjugation in
.) By the above proposition, many problems concerning Abelian extensions of exponent
of a field
can be reduced to the theory of Kummer extensions even if
. To be precise: If
is such an extension, then
is a Kummer extension, and its Kummer group is characterized by the condition: If
and
, then
, where
is a natural number which is defined modulo
by the condition
.
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 is trivial.
The theory of Kummer extensions carries over to the case of infinite Abelian extensions of exponent . When this is done, the Kummer pairing establishes a Pontryagin duality between the profinite group
(endowed with the Krull topology) and the discrete group
(see [1], [2]).
The theory of Kummer extensions, also known as Kummer theory, has an analogue in the case of extensions (1) with (Artin–Schreier theory). The role of the group
in that situation is played by the additive group of the prime subfield
of
. The main result of the theory is: Any Abelian extension
of exponent
of a field
is of the form
, where
are roots of equations of the type
(see [1]). There is also a generalization of this theory, due to E. Witt, to the case
, where
, using Witt vectors (cf. Witt vector).
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 .
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 |
Kummer extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kummer_extension&oldid=41887