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Results tagged with algebraic-number-theory
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user 101078
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
21
votes
Accepted
Brauer–Siegel's Theorem and application
If $K$ is a number field, then its regulator is always larger than $1/5$. In fact, the paper Friedman, E., Analytic formulas for the regulator of a number field, Invent. Math. 98 (1989) 599–622, conta …
- 7,193
4
votes
Accepted
Chowla's theorem on class number of real quadratic field
Note that
$$
\prod_{n=1}^{p-1}(1-\zeta^n)=\Phi_p(1)=p,
$$
where $\Phi_p(x)=\frac{x^p-1}{x-1}$ is the cyclotomic polynomial. Next, $1-\zeta^{p-n}=-\zeta^{-n}(1-\zeta^n)$, hence we have
$$
p=\prod_{n=1} …
- 7,193