The Arithmetic of Algebraic Numbers: An Elementary Approach
Chi-Kwong Li, David Lutzer · College Mathematics Journal · 2004
Let Q and R be the fields of rational and real numbers respectively. Recall that a real number r is algebraic over the rationals if there is a polynomial p with coefficients in Q that has r as a root, i.e., that has p(r) = 0. Any college freshman can understand that idea, but things get more challenging when one asks about arithmetic with algebraic numbers. For example, being the roots of x2 − 3 and x2 − 20 respectively, the real numbers r1 = √ 3 and s1 = 2 √ 5 are certainly algebraic over the rationals, but what about the numbers r1 + s1, r1s1 and r1? As it happens, all three are algebraic over the rationals. For s1 example, r1 +s1 is a root of x4 −46x2 +289. But how was that polynomial constructed, and what rationalcoefficient-polynomials have r1s1 and r1 as roots? Students who take a second modern algebra course will s1 learn to use field extension theory to show that the required polynomials must exist. They will learn that whenever r and s = 0 are algebraic over Q, then the field Q(r,s) is an extension of Q of finite degree with the consequence that r + s, rs and r s are indeed algebraic over Q (see [2, 3, 7]). However, one would hope