Intereting Posts

Prove $\sum_{k=0}^{58}\binom{2017+k}{58-k}\binom{2075-k}{k}=\sum_{k=0}^{29}\binom{4091-2k}{58-2k}$
Element of, subset of and empty sets
Evaluating $\sum_{n=1}^{\infty} \frac{4(-1)^n}{1-4n^2}$
Continuous functions on a compact set
What is the real life use of hyperbola?
Is the matrix square root uniformly continuous?
Uniform convergence, but no absolute uniform convergence
Find all real numbers $x$ for which $\frac{8^x+27^x}{12^x+18^x}=\frac76$
Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$
Does every irreducible representation of a compact group occur in tensor products of a faithful representation (and its dual)?
Is the Fibonacci constant $0.11235813213455…$ a normal number?
Proof Check Lemma 2.2.10 in Tao
$\operatorname{spectrum}(AB) = \operatorname{spectrum}(BA)$?
Are there any number $n$ such that $a_n = 0 \mod (2n + 1) $ where $a_0 = 1, a_1 = 4, a_{n + 2}=3 a_{n + 1} – a_{n}$?
How can I deduce $\cos\pi z=\prod_{n=0}^{\infty}(1-4z^2/(2n+1)^2)$?

I am starting to read about the Kronecker-Weber Theorem. It says that any abelian extension of $\mathbb{Q}$ is contained in a cyclotomic extension. I read somewhere that for quadratic extensions the proof is not very difficult.

Can anyone tell me any reference material for the proof of the kronecker weber theorem for quadratic extensions ?

- Finding the ring of integers of $\mathbb Q$ with $\alpha^5=2\alpha+2$.
- Elements of cyclotomic fields whose powers are rational
- Finite abelian groups as class groups
- $L$-function, easiest way to see the following sum?
- How to determine a Hilbert class field?
- Class number computation (cyclotomic field)

- Galois group of $K(\sqrta)$ over $K$
- Golden Number Theory
- Ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z$
- What is a good way to find an algebraic field extension that is not separable and not normal?
- $K$-monomorphism that is not $K$-automorphism?
- Every finite group is isomorphic to some Galois group for some finite normal extension of some field.
- Divisibility of discriminants in number field extensions
- Find the minimal polynomial of $\sqrt2 + \sqrt3 $ over $\mathbb Q$
- A certain inverse limit
- Field extensions, inverse limits, notation and roots of unity

Look up Gauss sums. They express $\sqrt{p}$ as a function of roots of unity. They occur in Gauss’s fourth proof of quadratic reciprocity. The classic textbook by Ireland and Rosen “A classical introduction to modern number theory” has extensive discussion of them and their generalisations. Gauss sums are really just Lagrange resolvents for the cyclotomic equation. Lang’s book “Algebraic Number Theory” also explains conceptually the relation between reciprocity and the fact that quadratic fields are subfields of cyclotomic fields. Weber’s original proof used this as one of his cases.

- Fermat's Last Theorem and Kummer's Objection
- Tensors as mutlilinear maps
- $\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$
- Integrate square of the log-sine integral: $\int_0^{\frac{\pi}{2}}\ln^{2}(\sin(x))dx$
- What does the factorial of a negative number signify?
- How to solve this Complex inequality system
- Please explain inequality $|x^{p}-y^{p}| \leq |x-y|^p$
- Prove that $h^{(k)}(0)=\lim_{t\to0}\frac{\sum_{j=0}^k\binom{k}{j}(-1)^{k-j}h(jt)}{t^k}$
- arbitrary large finite sums of an uncountable set.
- Compute: $\sum\limits_{n=1}^{\infty}\frac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)\cdot (2n+1)}$
- Continued Fraction
- Limit of an integral question: $\lim \limits _{h \to \infty} h \int \limits _0 ^\infty e ^{-hx} f(x) \, d x = f(0)$
- Is a sphere a closed set?
- Ordinals definable over $L_\kappa$
- Calculus 2 integral $\int {\frac{2}{x\sqrt{x+1}}}\, dx$