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Law of Quadratic Reciprocity Let p and q be odd primes. Then

\[ \left( \frac { p } { q } \right) \left( \frac { q } { p } \right) = ( - 1 ) ^ { \frac { p - 1 } { 2 } \frac { q - 1 } { 2 } } \iff \begin{cases} (\frac{p}{q})=(\frac{q}{p}) &\text{if either p or q is }1\bmod 4\\ (\frac{p}{q})=-(\frac{q}{p}) &\text{if both p and q are }3\bmod 4 \end{cases} \]

Gauss’s lemma Let \(p\) be an odd prime, \(q\) be an integer coprime to \(p\). Take the least residues of \( Q=\{q, 2q,\cdots,\frac{p-1}{2} q \} \), i.e. reduce them to integers in \( [0, p-1] \). Let \(u\) be the number of members in this set that are greater than \(p/2\). Then

\[ (\frac{q}{p})=(-1)^u \]

Euler’s Criterion Let p be an odd prime, and a an integer not divisible by p. Then

\[ (\frac{a}{p}) \equiv a^{\frac{p-1}{2}}\mod p \]

Primitive root theorem. Let p be a prime. Then for any d dividing \( p-1 \), there are exactly \( \phi(d) \) elements of order d in \( (\mathbb Z / p \mathbb Z)^\times \). In particular there are \( \phi(p-1) \) primitive roots mod p.

Fundamental Theorem of Algebra. Every polynomial of degree greater than zero with complex coefficients has at least one zero.

The Euler-Poincaré formula describes the relationship of the number of vertices, the number of edges and the number of faces of a manifold. It has been generalized to include potholes and holes that penetrate the solid.

The following are equivalent for a subgroup $H$ of a group $G$:

1. $H = C_G(H)$.
2. $H$ is an abelian subgroup of $G$ and $C_G(H) \le H$ (i.e., $H$ is a self-centralizing subgroup of $G$).
3. $H$ is an abelian subgroup of $G$ and it is not contained in any bigger abelian subgroup of $G$.