Kummer extension

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