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Results tagged with algebraic-number-theory
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user 430
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
7
votes
Accepted
Prime splitting in cubic field, congruence
I don't see why the restriction to galois extensions is necessary. Consider, for example, the non-galois cubic field $K = \mathbb Q(\sqrt[3]{2})$. Then no prime congruent to 2 mod 3 splits completely …
- 10.3k
12
votes
When can number rings be spanned (as $\mathbb{Z}$-modules) by units?
As was already pointed out, the only imaginary quadratic fields with $R=\mathcal O$ are $\mathbb Q(i)$ and $\mathbb Q(\zeta_3)$. This follows pretty easily from the structure of $\mathcal O^\times$. O …
- 10.3k
7
votes
Accepted
Exercise in Milne's CFT notes
How did you determine that the index $(\mathcal O_L : M)$ is $8$? It seems to me that it's actually $160$, which is divisible by $5$.
Incidentally, a quick way to see that Milne is correct is to note …
- 10.3k
5
votes
Accepted
Which formulae of Euler is Fröhlich referring to?
I believe the reference is to this formula of Euler (see here): If $P(x)/Q(x)$ is a rational function and $ax+b$ is a simple factor of $Q(x)$, then the coefficient of $1/(ax+b)$ in the partial fractio …
- 10.3k
7
votes
Accepted
Heegner Points and Binary Quadratic Forms
I think the following facts, which you can find in Cox's book Primes of the Form $x^2+ny^2$, will alleviate your confusion. First off, if ${\mathfrak a}=[\alpha,\beta]$ is a proper ideal of ${\mathcal …
- 10.3k