For $K = \mathbb{Q}(\sqrt{d})$ with $d$ square-free, a prime $p \nmid 2d$ splits, remains inert, or ramifies according to the Legendre symbol $\left(\frac{d}{p}\right) = 1, -1, 0$.

For $d = -1$ (Gaussian integers): primes $p \equiv 1 \pmod{4}$ split, $p \equiv 3 \pmod{4}$ stay inert, and $p = 2$ ramifies as $(2) = (1+i)^2$.

This connection between quadratic reciprocity and splitting behavior extends to higher reciprocity laws via class field theory.

If $K$ contains a primitive $n$-th root of unity $\zeta_n$ and $\alpha \in K$ with $\sqrt[n]{\alpha} \notin K$, then $L = K(\sqrt[n]{\alpha})$ is a Kummer extension.

Ramification in $L/K$ occurs precisely at primes dividing $(\alpha)$. If $\mathfrak{p}^e || (\alpha)$ (exact power), then $\mathfrak{p}$ ramifies with index dividing $\gcd(n, e)$.

For $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$: the prime $(2)$ ramifies, while all other primes are unramified.

The conductor of $\mathbb{Q}(\zeta_n)$ is $n$ (or $2n$ if $n$ is odd). This means primes $p | n$ ramify, all others are unramified.

For $\mathbb{Q}(\zeta_{12}) = \mathbb{Q}(\zeta_3, \zeta_4)$: primes 2 and 3 ramify. The factorizations are:

• $(2) = \mathfrak{p}_2^4$ with $f = 1, g = 1$
• $(3) = \mathfrak{p}_3^2$ with $f = 1, g = 1$

Other primes split according to their residue classes modulo 12.

The discriminant of $K = \mathbb{Q}(\alpha)$ where $\alpha$ satisfies $f(x) = 0$ can be computed from $f$ via: $\Delta_K = \pm N_{K/\mathbb{Q}}(f'(\alpha))$

when $\mathcal{O}_K = \mathbb{Z}[\alpha]$ (monogenic case).

For $K = \mathbb{Q}(\sqrt{d})$: $\Delta_K = d$ if $d \equiv 1 \pmod{4}$ and $\Delta_K = 4d$ otherwise. Primes dividing $\Delta_K$ are precisely the ramified primes.

For unramified primes $\mathfrak{p}$ in an abelian extension $L/K$, the Artin symbol $\left(\frac{L/K}{\mathfrak{p}}\right) \in \text{Gal}(L/K)$ is the Frobenius element.

This satisfies: $\left(\frac{L/K}{\mathfrak{p}}\right)(\alpha) \equiv \alpha^{N(\mathfrak{p})} \pmod{\mathfrak{P}}$ for all $\alpha \in \mathcal{O}_L$.

The Artin map $\mathfrak{p} \mapsto \left(\frac{L/K}{\mathfrak{p}}\right)$ extends to ideals and gives the Artin reciprocity isomorphism relating ideals to Galois groups.

The Hasse principle (local-global principle) states that certain properties hold globally if they hold locally everywhere.

For quadratic forms: $Q$ represents zero over $\mathbb{Q}$ if and only if it represents zero over $\mathbb{R}$ and all $\mathbb{Q}_p$.

Ramification determines which primes require checking. For $x^2 + y^2 - dz^2 = 0$: check $p = 2, p | d$, and $p = \infty$ (the real place).