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Understanding Euler's Criterion is fundamental when delving into the theory of quadratic residues. It is a statement in number theory that provides a clear criterion for determining whether a number is a quadratic residue modulo a prime. In simpler terms, it helps us figure out when a number, when squared, resembles another number after considering only the remainder (or 'modulus') when divided by a prime number.

In the context of our problem, Euler’s Criterion comes into play by setting up the equation \[(-1)^{\frac{q-1}{2}} \bmod q\]. If this is equivalent to 1, we can affirm that -1 is a quadratic residue modulo the prime number q. The connection to the exercise lies in proving that this relationship holds true precisely when the prime number q is one more than a multiple of 4, harnessing this powerful criterion to arrive at a clear and concise mathematical proof.

Prime numbers are like the building blocks in the world of integers: these numbers are greater than 1 and cannot be formed by multiplying two smaller natural numbers. A prime number can only be divided by 1 and itself without leaving a remainder. They are essential in number theory, and our problem specifically concerns prime numbers because the properties of quadratic residues and Euler's Criterion fully come into play in the context of primes.

When proving statements about quadratic residues, the prime nature of the number adds a layer of certainty and simplifies calculations in modular arithmetic – which is essential to our understanding and solution of the given problem.

Often referred to as clock arithmetic, modular arithmetic is a system that deals with integers and their remainders when divided by a fixed number (the modulus). It's a fundamental part of the equation in our problem involving \(q \bmod 4\) and Euler's Criterion.

Modular arithmetic, in essence, helps us manage numbers by how they relate in terms of their remainders when divided by a modulus. In the solution for the exercise, for instance, when we consider \(q \bmod 4\), we're interested in whether q, when divided by 4, leaves a remainder of 1. This branch of mathematics is not just fascinating but instrumental in several fields, including computer science, cryptography, and, of course, number theory.

Proofs are the bedrock of mathematical truth, providing the rigorous justifications needed to establish the validity of mathematical statements. Different proof techniques are used depending on the nature of the statement. In our exercise solution, we've used a direct proof technique, showing through logical deductions that if a prime number satisfies \(q \bmod 4 = 1\), then -1 is a quadratic residue.

Another aspect of mathematical proofs is the 'if and only if' format seen in the exercise. This requires proving the statement in both directions – that not only does \(q \bmod 4 = 1\) imply that -1 is a quadratic residue but also that if -1 is a quadratic residue, then \(q \bmod 4 = 1\). By nailing down both directions, we confirm the bi-conditional relationship, an elegant and compelling form of mathematical argument.