Kummer extension

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. $$ \mathrm{Gal}(K/k) \cong \mathrm{Hom}(A(K/k),\mu(n) $$

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 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