Show that there are p^2-p elements with multiplicative...

Show that there are $p^2-p$ elements with multiplicative inverses in $Z_{p^2}$ when $p$ is prime. If $x$ has a multiplicative inverse, what is $x^{p^2-p} \bmod p^2 ?$ Is the same statement true for an element without an inverse? (Working out an example might help here.) Can you find something that is true about $x^{p^2-p}$ when $x$ does not have an inverse?

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Key Concepts

Units in Modular Arithmetic

In the ring of integers modulo n, the units are the elements that have multiplicative inverses. For Z??²?, an element is a unit if and only if it is coprime with p²; since p is prime, this means the element is not divisible by p. This distinction divides the elements into those with and without multiplicative inverses.

Euler's Totient Function

Euler's totient function, denoted ?(n), counts the numbers up to n that are relatively prime to n. For a prime squared like p², the totient function evaluates to ?(p²) = p² - p, which exactly counts the number of units in Z??²?.

Euler's Theorem

Euler’s theorem states that if an integer x is coprime with n, then x raised to the power ?(n) is congruent to 1 modulo n. In the context of Z??²?, this implies that for any invertible element x, the congruence x^(p²-p) ? 1 (mod p²) holds.

Behavior of Non-Units

For elements in Z??²? that are not units (i.e., elements divisible by p), Euler’s theorem does not apply. Typically, if x is a multiple of p, then raising x to the power (p²-p) results in a number that is congruent to 0 modulo p², due to the accumulation of the factor p in the exponentiation process.

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