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# State/Eulers criterion

(Redirected from State/Euler Criterion)

Proposition: (Euler Criterion) Suppose that $a$ is an integer and $p$ is an odd prime number. Then,

• $\bar a^{(p-1)/2} = \bar 1$, mod $p$, if and only if $\bar a$ is a nonzero quadratic residue, mod $p$.
• $\bar a^{(p-1)/2} = \overline{-1}$, mod $p$, if and only if $\bar a$ is nonzero, and not a quadratic residue, mod $p$.
• $\bar a^{(p-1)/2} = \bar 0$, mod $p$, if and only if $\bar a = \bar 0$, i.e., $a$ is a multiple of $p$.

## Logical Connections

This statement logically relies on the following definitions and statements: Def/Quadratic residue, Def/Residue, State/Root counting over a field, State/Multiplicative inverses exist mod p, State/No nonzero nilpotents in a field, State/Half of nonzero residues are quadratic residues, State/Fermats little theorem

The following statements and definitions rely on the material of this page: Def/Legendre symbol

To visualize the logical connections between this statements and other items of mathematical knowledge, you can visit the following cluster(s), and click the "Visualize" tab: Clust/Theory of quadratic residues