# Questions tagged [local-field]

For questions about local field, which is a special type of field that is a locally compact topological field with respect to a non-discrete topology.

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### Formal groups over discrete valuation rings

Let $k/\mathbb{Q}_p$ be a finite extension. Let $k^{ur}$ be its maximal unratified extension and consider $K$ its completion. Let $\varphi$ be the Frobenius morphism in Gal$(k^{ur}/k)$ and whit the ...
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### Absolute Galois group of the p-adic completion of a valuation field

Let $K$ over $\mathbf{Q}_p$ be an algebraic extension, not complete for $|\cdot|_K$ (the unique extension of $p$-adic norm $|\cdot|_p$ to $K$). (For example, $K=\cup_n\mathbf{Q}_p(\zeta_{p^n})$ the ...
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### On the concept of primary element

Let $\ell$ be an odd prime number,$\zeta_{\ell}:=e^{2\pi i/\ell}$, $F:=\mathbb{Q}(\zeta_{\ell})$ be a cyclotomic field, $\mathcal{O}_F$ be its integer ring and $\lambda:=1-\zeta_{\ell}$. [Ireland-...
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### What is the decomposition of global units $1+\mathfrak{p}$?

Let $p \geq 2$ be prime and $K=\mathbb Q(\zeta_p),~\zeta^{p}=1$ with ring of integers $\mathcal{O}_K$. we denote by $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ be ...
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### Tate cohomology of units

Show that for any local field K, there is a finite Galois extension L, such that $H^{i}_{T}(Gal(L/K),O_L^*)$ does not vanish for all i, here $O_L$ is ring of algebraic integers of L. I only know that ...
1 vote
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### uniqueness of local Artin map

This problem defines a map with the same properties as local Artin map and asks you to prove they are equal. I'm having problems with b) and c). Is the first part of b) comes from Galois ...
1 vote
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### Why's $v_L (x)=\frac{1}{n}v_K(N_{L/K}(x))$ integer for unramified local field extension $L/K$?

If $K$ is a local field and $L/K$ is a finite extension, then the valuation $v_K$ can be extended uniquely to a valuation $v_L$ of $L$ such that $v_L$ restricted to $K$ is equal to $v_K$. This is one ...
1 vote
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### What is the quotient group $\mathfrak{q}^2/\mathfrak{p}^2\mathbb Z_p$?

Let $p \geq 2$ be prime and $K=\mathbb Q(\zeta_p),~\zeta^{p}=1$ with ring of integers $\mathcal{O}_K$. we denote by $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ be ...
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### Discriminant of a $V_4$-extension of local fields is the product of discriminants of intermediate fields

Let $L/K$ be a Galois extension of $p$-adic fields with Galois group $V_4 = C_2 \times C_2$, and write $d_{L/K}$ for its discriminant, which is an ideal of $\mathcal{O}_K$. The extension $L/K$ has ...
1 vote
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### Example of rational jumps in upper ramification filtration

Can anyone point to me or write down an explicit example of a non-integer jump of the higher ramification filtration in positive characteristic and the corresponding equations of the intermediate ...
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### Do formal Laurent series of norm 1 over a local field come from power series over ring of integers?

Let $K/\mathbb Q_p$ and $L/K$ be finite extensions of fields, with $O_K$ resp. $O_L$ being the rings of integers in $K$ resp. $L$. Let $N:L\to K$ denote the norm map from $L$ to $K$. Consider the ...
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### JS Milne Algebraic Number Theory 8.6 : normalized absolute values for local fields

This is from JS Milne's notes on Algebraic Number Theory, lemma 8.6 Let $K$ be a local field with normalized absolute value $|.|_K$. Let $L$ be a finite separable extension of degree $n$, and $|.|_L$ ...
### Norm of a root of unity $\zeta$ is $1$ only if $\zeta=1$?
Let $p$ be an odd prime number. Let $K/\mathbb Q_p$ be a finite extension, with $K$ having residue field $\mathbb F_q$ of order $q$ some power of $p$. Let $\zeta\in\mu_{q-1}\subset K$ be a $(q-1)$st ...