By Theo Bühler

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The set Œa is called the residue class or congruence class of a modulo n. a/ the unique number r such that 0 Ä r < n and a r is divisible by n. 7). a/ is the unique element of Œa that lies in the interval f0; 1; : : : ; n 1g. a/ is the label on the circumference of the clock face at the point where a falls when the number line is wrapped around the clock face. 3. mod n/. (b) Œa D Œb. b/. (d) Œa \ Œb ¤ ;. Proof. mod n/. mod n/, by the previous lemma, part (c). This shows that Œa Â Œb. mod n/, so Œb Â Œa.

How do we actually compute the prime factorization of a natural number? The conceptually clear but computationally difficult method that you learned in school forpfactoring a natural number n is to test all natural numbers no larger than n to see if any divides n. If no factor is found, then n must be prime. n=a/ and proceed to search for factors of a and n=a. We continue this procedure until only prime factors appear. Unfortunately, this procedure is very inefficient. Better methods are known, but no truly efficient methods are available for factoring very large natural numbers.

For part (a), if uv D 1, then both of u; v must be nonzero. If either of u or v had positive degree, then uv would also have positive degree. Hence both u and v must be elements of K. 1 uv/ D 0. If g D 0, then k D 0 meets the requirement. Otherwise, 1 uv D 0, so both u and v are elements of K, by part (a), and k D v satisfies the requirement. The remaining parts are left to the reader. ■ What polynomials should be considered the analogues of prime numbers? The polynomial analogue of a prime number should be a polynomial that does not admit any nontrivial factorization.