The uniqueness of the power series representation of certain fields with valuations. (English) Zbl 0019.04901

MacLane, Saunders

Ann. Math. (2) 39, 370-382 (1938).

Let \(K=K^{(m)}\) be a field on which a discrete valuation \(V\) of rank \(m\) is defined. The valuation \(V\) determines a series of \(m\) homomorphisms \(H^{(i)}\) \((i = 1,\ldots, m)\) given as \[ H^{(m)}K^{(m)} = \{K^{(m-1)}, \infty\},\quad H^{(m-1)}K^{(m-1)}= \{K^{(m-1)}, \infty\}, \ldots , H^{(1)}K^{(1)}=\{\mathfrak K, \infty\}, \]
where \(\mathfrak K\) denotes the field of residue classes of \(K\) with respect to \(V\). Each homomorphism \(H^{(i)}\) determines a discrete valuation of rank 1 on \(K^{(i)}\). A field \(K\) is called “step perfect” with respect to \(V\) if each field \(K^{(i)}\) is perfect with respect to the valuation \(H^{(i)}\). It is shown that each field \(K\) of the type described possesses an immediate extension \(L\) which is step perfect, i. e. the value group of \(L\) coincides with the value group of \(K\) and the respective fields of residue classes coincide, too. Let \(\chi^{(i)}\) be the characteristic of \(K^{(i)}\). Then there exists a unique step perfect extension \(\overline K\) of \(K\) (to within analytic isomorphisms) provided that \(\chi^{(m)} = \ldots = \chi^{(1)}= 0\).
The proof of this theorem essentially depends an the following lemma: “If \(HK = \{\mathfrak K, \infty\}\) is a discrete perfect homomorphism of rank 1, if \(N\subset K\) is a subfield such that \(H(N^*)\subset \mathfrak K^*\), and if \(\mathfrak K\) is separable over \(H(N)\) then there exists a subfield \(M\) of \(K\) with \(N\subset M\) and \(H(M^*)= \mathfrak K^*\). (\(N^*\), \(\mathfrak K^*\) denote the multiplicative groups of the fields \(N\), \(\mathfrak K\) respectively.)
Finally, the author shows that in general the uniqueness theorem breaks down if the condition concerning the characteristics does not hold, i. e. if \(\chi^{(i)}\neq 0\).

Reviewer: O. F. G. Schilling (Baltimore)

MSC:

12J20

General valuation theory for fields

Keywords:

discrete valuation; step perfect field