Summary: First Steps With General Congruences
Although we cannot as easily fully solve more general congruences than linear ones, there are many useful and elementary results to explore.
- As a prelude, we explore Question 7.1.1 about when we have square roots of \(\pm 1\text{,}\) modulo \(n\text{.}\)
- Can we use some of the methods from linear congruences for polynomials?
- We can combine solutions to polynomials in a similar way to the Chinese Remainder Theorem (Fact 7.2.2).
- In Hensel’s Lemma we see how to use a solution modulo a prime power to create a solution modulo a higher power of the same prime.
- A key approach in solving congruences is to remember that the nature of the solutions may also be expressed in terms of a congruence. Fact 7.3.2 is a first good example of this, giving a complete analysis of square roots of one.
- We next see in Lagrange’s Theorem for Polynomials that when our modulus is prime, solutions of polynomials are limited more closely by our previous experience.
- Two towering theorems giving theoretical tools to harness more complex congruences are Wilson’s Theorem and Fermat’s Little Theorem.
- Finally, we explore Mordell curves again in an effort to motivate a deeper understanding of Epilogue: Why Congruences Matter.
The Exercises focus on polynomial congruences, but include a little practice of Fermat’s Little Theorem. After this we have alternate combinatorial proofs provided of Fermat’s Little Theorem and Wilson’s Theorem; see especially the section on Counting motivation.
