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Extra resources for An Introduction to p-adic Numbers and p-adic Analysis [Lecture notes]

Example text

We will not prove the next result, the interested reader should consult [5]. 2. There exists a field Qalg p containing Qp as a subfield and having the following properties: (a) every α ∈ Qalg p is algebraic over Qp ; alg (b) every polynomial f (X) ∈ Qalg p [X] has a root in Qp . Moreover, the norm | |p on Qp extends to a unique non-Archimedean norm N on Qalg p satisfying N (α) = |α|p whenever α ∈ Qp . This extension is given by N (α) = minQp ,α (0) 1/d p , where d = degQp (α) = deg minQp ,α (X) is the degree of the minimal polynomial of α over Qp .

Let us look at some elements of Qalg p . Many examples can be found using the next two results. 3. Let r = a/b be a positive rational number where a, b are coprime. Then the polynomial X b − pa ∈ Qp [X] is irreducible over Qp and each of its roots α ∈ Qalg p has norm |α|p = p−a/b . Proof. This is a special case of [6, VIII theorem 16]. 4. If r = a/b is not an integer, then none of the roots of X b − pa in Qalg p are in Qp . 47 Proof. We have |α|p = p−a/b which is not an integral power of p. But from Chapter 2 we know that all elements of Qp have norms which are integral powers of p, hence α ∈ / Qp .

In real analysis, there are series which converge but are not absolutely convergent. For ∑ ∑ example, the series (−1)n /n converges to − ln 2 but 1/n diverges. Our next result shows that this cannot happen in Qp . 4. The series αn in Qp converges if and only if (αn ) is a null sequence. ∑ Proof. 1 the sequence of partial sums (sn ) is Cauchy since sn+1 − sn = αn is a null sequence. 1 we see that the sequence (sn ) is Cauchy and hence converges. ∑ So to check convergence of a series αn in Qp it suffices to investigate whether lim(p) αn n→∞ = 0.

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