2
votes
Accepted
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.
...
- 159k
2
votes
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 ...
- 43.2k
1
vote
Accepted
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
1
vote
Accepted
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
1
vote
Accepted
Given $F_0\subset\cdots\subset F_n$ and intermediate $E$, finding $F=E_0\subset\cdots\subset E_m=E$ such that $\max[E_i:E_{i-1}]\le\max[F_i:F_{i-1}]$
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\...
- 13.3k
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