Abstract
In this chapter, we review some facts from the theory of non-Archimedean fields that will be used in later chapters. We discuss completions, Hensel’s lemma, the Newton polygon method, extensions of local fields, ramification, and valuations and completions of global function fields. We also discuss some basic notions of analysis in the setting of complete non-Archimedean fields, such as the radius of convergence of a power series, the Weierstrass factorization theorem, and the existence and distribution of zeros of entire functions.
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Notes
- 1.
This condition is obviously necessary. Conversely, if G is not discrete, then there is a Cauchy sequence g1, g2, … with gn ∈ G. In that case, the sequence {gm − gm+1} consists of elements of G converging to 0.
- 2.
It is easy to show that \(\left |\cdot \right |\sim \left |\cdot \right |'\) if and only if \(\left |\cdot \right |\), and \(\left |\cdot \right |'\) define the same topology on K, i.e., if and only if K has the same open sets with respect to \(\left |\cdot \right |\) and \(\left |\cdot \right |'\).
- 3.
Although Hensel’s Lemma will be proved in a later section, its proof does not use the isomorphism \(\widehat {K}\cong k\!\left (\mkern -4mu\left (\pi \right )\mkern -4mu\right )\), so the argument here is not circular.
- 4.
This was originally proved by Tate.
- 5.
This is an idea that goes back to Isaac Newton; cf. [Gou20, p. 91].
- 6.
For Hensel’s Lemma to apply, instead of working over the completion \(\widehat {K}\) of a given field K equipped with a nontrivial non-Archimedean absolute value, it is enough to pass to the smaller field \(\widehat {K}\cap K^{\mathrm {sep}}\), called the Henselization of K. The ability to work over the Henselization, rather than completion, is important in some arithmetic problems, although it is not essential for our purposes.
- 7.
If K(α)∕K is Galois, then K(α) = L, so n = [K(α) : K] = [L : Kn][Kn : K] = n[Kn : K], which implies Kn = K.
- 8.
If L∕K is abelian, then every intermediate field K ⊂ K′⊂ L is Galois over K. On the other hand, K(α) is such a field.
- 9.
This is a non-Archimedean phenomenon: the series \(\sum _{n\geq 1}\frac {(-1)^n}{n} x^n\in \mathbb {R}\!\llbracket x\rrbracket \) converges for |x| < 1 and x = 1 but diverges for x = −1.
- 10.
This is a special property of entire functions over non-Archimedean fields since both \(\exp (x)=\sum _{n\geq 0} x^n/n!\) and 1 have empty sets of zeros over \(\mathbb {C}\).
- 11.
But note that ∘ is not commutative, and f ∘ (g + h) is not always equal to f ∘ g + f ∘ h.
- 12.
Note that this property characterizes \( \operatorname {\mathrm {Frob}}_{\mathfrak {P}}\) uniquely also as an element of \( \operatorname {\mathrm {Gal}}(L/K)\). Indeed, if \(\sigma \in \operatorname {\mathrm {Gal}}(L/K)\) has the property that \(\sigma (\alpha )\equiv \alpha ^{\# k}\ (\mathrm {mod}\ \mathfrak {P})\) for all α ∈ B′, then for \(\alpha \in \mathfrak {P}\) we have \(\sigma (\alpha )\equiv 0 \ (\mathrm {mod}\ \mathfrak {P})\), so \(\sigma (\alpha )\in \mathfrak {P}\). It follows that \(\sigma (\mathfrak {P})=\mathfrak {P}\), and therefore, \(\sigma \in D(\mathfrak {P}/\mathfrak {p})\).
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Papikian, M. (2023). Non-Archimedean Fields. In: Drinfeld Modules. Graduate Texts in Mathematics, vol 296. Springer, Cham. https://doi.org/10.1007/978-3-031-19707-9_2
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