Show that if p is an odd prime and a is an integer not...

Show that if $p$ is an odd prime and $a$ is an integer not divisible by $p$ , then the congruence $x^{2} \equiv a(\bmod p)$ has either no solutions or exactly two incongruent solutions modulo $p .$

Key Concepts

Finite Field Structure

When working modulo an odd prime, the set of integers modulo that prime forms a finite field. In a finite field, every nonzero element has a multiplicative inverse and the usual algebraic rules apply. This property guarantees that any nonconstant polynomial, such as a quadratic, has at most as many distinct solutions as its degree, which is crucial in analyzing congruence equations.

Quadratic Congruence

A quadratic congruence is an equation of the form x² ? a (mod m) where one seeks integers x that satisfy this condition modulo m. In the context of a finite field (when m is prime), the quadratic congruence becomes a polynomial equation of degree two. By the fundamental theorem of algebra (adapted for finite fields), such an equation can have at most two distinct solutions, and if one solution is found, its negative modulo the prime provides the other distinct solution.

Quadratic Residue

An integer a is termed a quadratic residue modulo a prime if there exists an integer x such that x² is congruent to a modulo that prime. The concept of quadratic residues establishes a criterion for the solvability of quadratic congruences; if a is a quadratic residue, then the equation has exactly two distinct solutions, and if not, the equation has no solution. This dichotomy is essential in understanding the nature of quadratic equations in modular arithmetic.

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