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A First Course in Abstract Algebra: Rings, Groups, and by Marlow Anderson

By Marlow Anderson

A part of the PWS complicated arithmetic sequence, this article comprises chapters on polynomials and factoring, precise factorization, ring homomorphisms and beliefs, and constructibility difficulties and box extensions.

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Sample text

To actually determine gcd(a, b) we would rather not check all the possibilities less than |a| and |b|. Fortunately, we don’t have to, because there is an algorithm that determines the gcd quite efficiently. This first appears as Proposition 2 of Book 7 of Euclid’s Elements and depends on repeated applications of the Division Theorem; we call it Euclid’s Algorithm. 2 Suppose that a, b, q, r are integers and b = aq + r. Then gcd(b, a) = gcd(a, r). Proof: To show this, we need only check that every common divisor of b and a is a common divisor of a and r, and vice versa, for then the greatest element of this set will be both gcd(b, a) and gcd(a, r).

We now consider the set of all residue classes modulo m. We denote this set by Zm . That is, Zm = {[0], [1], [2], · · · , [m − 1]}. Be careful to note that we are considering here a set of sets: Each element of the finite set Zm is in fact an infinite set of the form [k]. While this construct seems abstract, you should take heart from the fact that for the most part, we can focus our attention on particular representatives of the residue classes, rather than on the entire set. 2 Arithmetic on the Residue Classes We are now ready to define an ‘arithmetic’ on Zm which is directly analogous to (and indeed inherited from) the arithmetic on Z.

We are thus allowing the always possible ‘trivial’ factorizations p = (1)(p) = (−1)(−p). We are not allowing ±1 to be irreducible because it would unnecessarily complicate the formal statement of the Fundamental Theorem of Arithmetic that we make below. Because 0 = (a)(0) for any integer a, it is clear that 0 is not irreducible. Finally, notice that if p is irreducible, then so is −p. This means that in the arguments that follow we can often assume that p is positive. The positive integers that are irreducible form a familiar list: 2, 3, 5, 7, 11, 13, 17, · · · .

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