Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$

My favorite way to prove this is to explicitly write down the quadratic Gauss sum: $$g_p = \sum_{a \in \mathbb F_p} \left( \frac{a}{p} \right) \zeta_p^{a}$$ Then you can show $g_p^2 = (-1)^{\frac{p-1}{2}} p$ by direct manipulation. This gives the result very explicitly!

See this exercise sheet for a more or less guided solution. Re uniqueness: what is the Galois group of the cyclotomic field? What does the Galois correspondence tell you?

The set of functions which map convergent series to convergent series

Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times

$G$ is non abelian simple group of order $<100$ then $G\cong A_5$

A lady and a monster

"If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer"

Integral: $\int_0^{\pi/12} \ln(\tan x)\,dx$

Intuition behind the Definition of Conditional Probability (for 2 Events)

The composition of two convex functions is convex

Intuitive explanation of the Fundamental Theorem of Linear Algebra

What is the second principle of finite induction?

Product of connected spaces

Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.