# Tagged Questions

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Let $f$ be: $$f(x)=\sum_{n=0}^{\infty} a_nx^n$$ Such that $a_n \in \mathbb{Z_p}$ and $a_n \to 0$ We denote $|f|=\max|a_n| \space$and $\deg(f)=\max\{n:|a_n|=|f|\}$. ($a$) Prove that if ...
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### Examples where there is no power integral basis

Let $A$ be a completion discrete valuation ring with quotient field $K$, $L/K$ finite and separable, and $B$ the integral closure of $A$ in $L$. Let $P, \mathfrak P$ be the unique maximal ideals of ...
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### Frobenius action on $\overline{\mathbb Q_p}$

Let $p$ be a prime number and let $F_p$ be the Frobenius automorphism of $\overline{\mathbb F_p}$. Given an explicit element $x$ of $\overline{\mathbb Q_p}$, how do I compute $F_p(x)$? Does it even ...
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### Is $f(x)=x^{4}-2x^2 +3$ Eiseinstein in 2-adic $\mathbb{Q}_{2}$?

I think it is because $|1|_{2}=1$, $|2|_{2}=2^{-1}\leq 1$ and $|3|$,where 3 is prime I am using the following Eisenstein criterion for $f(x)=a_{n}x^{n}+...+a_{0}$: $|a_{n}|=1$, $|a_{0}|=prime$ and ...
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### Possible Quadratic extensions of $\mathbb{Q}_{2}$ are 1-1 with $\mathbb{Q}_{2}^{*}/(\mathbb{Q}_{2}^{*})^{2}$

Why do they have to be units? $\mathbb{Q}_{2}[\sqrt{2}]$ is a quadratic extension but $|2|_{2}=\frac{1}{2}\neq 1$. Where do I need them to be units below? ...
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### Degree of closure of $\mathbb{Q}_p$

In order to prove that algebraic closure of $\mathbb{Q}_p$ is infinite, I took the polynomial $x^n-p$ with $n>1$ over $\mathbb{Q}_p$ to show that this eqaution has no solution for infinite cases to ...
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### $p$-part of cyclotomic character

Let $K$ be a number field, $\bar K$ a separable closure and $p$ a rational prime and assume that $p \not= char(K)$. Consider the extension $$K(\mu_{p^\infty}) \mid K$$ which is obtained by adjoining ...
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### Number of field extensions of $\mathbb{Q}_p$

If I know the index $(\mathbb{Q}_p^{\times} : (\mathbb{Q}_p^{\times})^n)$ for some $n \in \mathbb{N}$, is it possible to know how many field extensions of $\mathbb{Q}_p$ of degree $n$ there are? This ...
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### Embedding into $p$-adic complex numbers

As I'm reading notes about the Leopoldt conjecture, the following question came to my mind: Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of the algebraic closure of the ...
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### Properties of squares in $\mathbb Q_p$

Let $\mathbb Q_p$ be the field of $p$-adic numbers. I know that for $p \neq 2$ an element $x=p^n u \in \mathbb Q_p^\times$ (with $n \in \mathbb Z$ and $u \in \mathbb Z_p^\times$) is a square if and ...
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### Discrete valuation on $p$-adic numbers

For the ring of $p$-adic integers $\mathbb Z_p$ let $\mu_n: \mathbb Z_p \to \mathbb Z / p^n \mathbb Z$ be the projection mapping. Consider $\mathbb Z / p^n \mathbb Z$ with the discrete topology. Is ...
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### For what primes $p$ does the series $1!+2!+3!+4!+ \cdots$ converge $p$-adically?

this is a question from a book I'm struggling with, please could you provide a clear proof For what primes p does the series $1!+2!+3!+4!+ \cdots$ converge $p$-adically? kind thanks
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### For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically

For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically and, when it does, to what limit?
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### Show that the ﬁeld of p-adic numbers is complete

this is a question from a book I'm struggling with, please could you provide a clear proof Show that the ﬁeld of p-adic numbers is complete i.e. that a sequence of p-adic numbers converges if and ...
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### Why $p$-adically interpolate?

I'm studying $p$-adic analysis now and particularly $p$-adic interpolation; for example, constructions like $p$-adic $L$-functions (Kubota-Leopoldt style). I'm having some difficulty though, and I'd ...
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### Necessary and sufficient conditions for Hensel lifting in the multidimensional case

in Multidimensional Hensel lifting, @Hurkyl gave a neat sufficient condition for the existence of $p$-adic liftings in the multidimensional case. I have finally gotten around (but please also see ...
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I asked a question the other day: Multidimensional Hensel lifting which @Hurkyl kindly and very elegantly answered. A follow-on from this is that I have tried to implement exactly the "algorithm" ...
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### Multidimensional Hensel lifting

I have a question about a practical application of (some) generalised form of Hensel's Lemma. I cannot find it stated in an appropriate form in Bourbaki or anywhere else, so here goes ... Let $p$ be ...
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### What is the index of the $p$-th power of $\mathbb Q_p$ in $\mathbb Q_p$

In this book it is listed as an exercise to compute the index $[\mathbb Q_p:\mathbb Q_p^p]$. This exercise is appended to a section concerning the structure of unit-group filters, investigating some ...
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### Set of locations where the Hilbert symbol is not equal to $1$

Let $V$ be the set of prime together with the symbol $\infty$. For a prime $v=p$, denote the $p$-adic numbers by $\mathbb{Q}_p$ and the real numbers by $\mathbb{Q}_\infty$. For $v\in V$ the Hilbert ...
I'm a little stuck with the proof of a theorem I'm trying to understand. The theorem is as follows: "For odd prime $p$, suppose for $\alpha \in Q_{p}$ (the p-adic rationals) that $|\alpha|_p=1$. Then ...
I'm reading a paper that purports to prove the proposition: Let $K/E$ be a cyclic extension of CM number fields of degree p (an odd prime number). Let $G$ be the Galois group. Let $t$ be the number ...