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# State/Multiplicative inverses exist mod p

Proposition: (Multiplicative inverses exist mod p) Suppose that $p$ is a prime number. If $\bar a$ is a nonzero residue, mod $p$, then $\bar a$ is an invertible residue, mod $p$.

In other words, the totient of $p$ is $p-1$: $\phi(p) = p-1$; there are $p$ residues, mod $p$, of which all but zero are invertible.

In other words, $\FF_p$ is a field.

## Logical Connections

This statement logically relies on the following definitions and statements: Def/Residue, Def/Invertible residue, State/Nonmultiples of a prime are relatively prime to the prime, State/Relative primality to the modulus is equivalent to invertibility of a residue

The following statements and definitions rely on the material of this page: State/Eulers criterion, State/Fermats little theorem, State/There are no zero divisors mod p, and State/Zolotarevs lemma

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/Modular arithmetic