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### Why is coboundary $\sigma \to \sigma m-m$ automatically continuous?

We don't need that $M$ is discrete, it can be any abelian topological group. The map is continuous since it is built from continuous maps. This phrase can be made precise by at least two methods. ...

### Norm is multiplicative?

Localize. Prove the norm map on ideals commutes with localization at prime ideals:  {\rm N}_{\mathcal O_L/\mathcal O_K}(I)\mathcal O_{K,\mathfrak p} = {\rm N}_{\mathcal O_{L,\mathfrak p}/\mathcal ...
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### Confused about lemma in Ivan Niven's Irrational Numbers about conjugate elements in a number field

For your first question: algebraic elements in general can have minimial polynomials of large degree, but the algebraic elements in this degree-$h$ field extension all have minimial polynomials of ...
• 74.9k
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### Suppose $0\to A\to B\to C\to D\to 0$ is exact. Let $0\to A[2]\to B[2]\to C[2]\to X\to 0$ be an exact sequence. Then, $X [2]$ is finite.

This is a bit ugly but works anyway. Consider the following diagram: $\require{AMScd}$ \begin{CD} @.B[2] @>>> C[2] @>>> X @>>> 0\\ @. @V{f}VV @V{g}VV @V{h}VV \\ ...
• 13.3k
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No. Consider $F_0 \subseteq F_1 \subseteq F_2 \subseteq F_3$ where $\mathrm{Gal}(F_3/F_0) \cong A_4$ and the subfields $F_0, F_1, F_2, F_3$ correspond to the subgroups $A_4, \mathbb{Z}/2\mathbb{Z} \... • 8,911 1 vote ### Twist of Elliptic curve$E_D: Dy^2=x^3+ax+b$is not isomorphic to$E:y^2=x^3+ax+b$? In general they are not isomorphic. Here is an argument I think should work. Suppose there were an isomorphism$\phi:E\to E_D$defined over$K$. Now take your isomorphism over$K(\sqrt{D})$and ... • 1,317 1 vote ### Why do we fix an extension of valuation of$K$to$\overline{K}$? The extension is not unique unless$K$is complete with respec to$v$. Take a global field$K$, with$v$being the valuation for a maximal ideal$\mathfrak m$, then for any algebraic extension$K\...
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