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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
56
votes
Accepted
Minimal polynomial of cos(π/n)
The minimal polynomial of $\cos(2\pi/n)$ (by William Watkins and Joel Zeitlin, The American Mathematical Monthly
Vol. 100, No. 5 (May, 1993), pp. 471-474) has full clarity on this matter (just take th …
13
votes
linear independence of $\sin(k \pi / m)$
We have
$$\sin\frac{\pi}{9}+\sin\frac{2\pi}9-\sin\frac{4\pi}9=\sin\frac{2\pi}{18}+\sin\frac{4\pi}{18}-\sin\frac{8\pi}{18}=\sin\frac{2\pi}{18}-\sin\frac{8\pi}{18}+\sin\frac{14\pi}{18},$$
and the latte …
1
vote
How do I find abelian cubic extension over $\mathbb{Q}$ with class number more than 1?
The earliest result I know which leads to many examples is that of Uchida:
Uchida, K.:
Class numbers of cubic cyclic fields.
J. Math. Soc. Japan 26(3): 447-453 (July, 1974)
See also Washington, L. …