290 VI Hensel’sp-adic Numbers
whereπis a generating element for the principal idealM,αn∈Sandαn=0forat
most finitely manyn<0. The map
a′=∑
n∈Zαntn→a=∑
n∈Zαnπnis a bijection of the fieldK((t))ontoF.SinceSis closed under addition this map pre-
serves sums, and sinceSis also closed under multiplication it also preserves products.
Finally, ifNis the least integer such thatαN=0, then|a|=|π|Nand|a′|=ρ−N
for some fixedρ>1. Hence the map is an isomorphism of the valued fieldK((t))
ontoF.
7 FurtherRemarks
Valued fields are discussed in more detail in the books of Cassels [1], Endler [3] and
Ribenboim [5].
For still more forms of Hensel’s lemma, see Ribenboim [6]. There are also gen-
eralizations to polynomials in several variables and to power series. The algorithmic
implementation of Hensel’s lemma is studied in von zur Gathen [4]. Newton’s method
for finding real or complex zeros is discussed in Stoer and Bulirsch [7], for example.
Proposition 20 continues to hold if the word ‘complete’ is omitted from its state-
ment. However, the formula given in the proof of Proposition 20 defines an absolute
value onEif and only if there is auniqueextension of the absolute value onFto an
absolute value onE; see Viswanathan [8].
Ostrowski’s Theorem 22 has been generalized by weakening the requirement
|ab|=|a||b|to|ab|≤|a||b|. Mazur (1938) proved that the only normed associative
division algebras overRareR,CandH, and that the only normed associative division
algebra overCisCitself. An elegant functional-analytic proof of the latter result was
given by Gelfand (1941). See Chapter 8 (by Koecher and Remmert) of Ebbinghaus
et al.[2].
8 SelectedReferences
[1] J.W.S. Cassels,Local fields, Cambridge University Press, 1986.
[2] H.-D. Ebbinghauset al.,Numbers, English transl. of 2nd German ed. by H.L.S. Orde,
Springer-Verlag, New York, 1990.
[3] O. Endler,Valuation theory, Springer-Verlag, Berlin, 1972.
[4] J. von zur Gathen, Hensel and Newton methods in valuation rings,Math. Comp. 42 (1984),
637–661.
[5] P. Ribenboim,The theory of classical valuations, Springer-Verlag, New York, 1999.
[6] P. Ribenboim, Equivalent forms of Hensel’s lemma,Exposition. Math. 3 (1985), 3–24.
[7] J. Stoer and R. Bulirsch,Introduction to numerical analysis, 3rd ed. (English transl.),
Springer-Verlag, New York, 2002.
[8] T.M. Viswanathan, A characterisation of Henselian valuations via the norm,Bol. Soc.
Brasil. Mat. 4 (1973), 51–53.