The Existence of a Multiplicative Inverse
Consider the field, Zp where p is a prime number. That is, consider
the numbers 0, 1, . . . , p − 1 and define arithmetic operations using modulo
arithmetic with modulus p. We would like to show that every number has a
multiplicative inverse.
First let’s consider a couple of examples.
2 · 3 ≡ 1 mod 5
5 · 7 ≡ 1 mod 17
Now let’s prove our claim. First note that when multiplies any two nonzero numbers modulo a prime, the produc is non-zero. This is so because for
a number to be zero modulo p requires that p be a factor of the number. But
in the case of a prime, the only way the prime can be a factor of the number
is if it was a factor of at least one of the multipicands—which would mean
that that number was zero. As we started off assuming that neither of the
multiplicands was zero, the product cannot be zero.
Now let us consider the product of k by two different numbers, n and
m. We find that kn − km ≡ k(n − m)modp. As neither of these numbers is
zero, the product is not zero either. That proves that each of the products
of k with another of the numbers 0, . . . , p − 1 is a different number. As we
have taken p products and their are p numbers in our number system, one
and only one of the products must equal one. That is, the number k has a
multiplicative inverse. QED.
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