5 The law of quadratic reciprocity

Theorem 5.1 Fermat’s little theorem

For \(a \not\equiv 0 \mod p\),

\[ a^{p - 1} \equiv 1 \mod p \]

Proof ▶

TODO

Theorem 5.2 Euler’s criterion

For \(a \not\equiv 0 (\mod p)\),

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

Proof ▶

TODO

Theorem 5.3 Product Rule

\[ (\frac{ab}{p}) = (\frac{a}{p}) \cdot (\frac{b}{p}) \]

Proof ▶

TODO

Theorem 5.4 Lemma of Gauss

TODO

Proof ▶

TODO

Theorem 5.5 Quadratic reciprocity I

TODO

Proof ▶

TODO

Theorem 5.6

The multiplicative group of a finite field is cyclic

Proof ▶

TODO

Theorem 5.7 A

TODO

Proof ▶

TODO

Theorem 5.8 B

TODO

Proof ▶

TODO

Theorem 5.9 Quadratic reciprocity II

TODO

Proof ▶

TODO