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- W2321007863 abstract "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 s1 ? As it happens, all three are algebraic over the rationals. For example, r1 +s1 is a root of x4−46x2 +289. But how was that polynomial constructed, and what rationalcoefficient-polynomials have r1s1 and r1 s1 as roots? Students who take a second modern algebra course will learn to use field extension theory to show that the required polynomials must exist. They will learn that whenever r and s 6= 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 s are indeed algebraic over Q (see [2, 3, 7]). However, one would hope that students would encounter more elementary solutions for such basic arithmetic questions. Furthermore, one might want to know how to construct rational-coefficient polynomials that have r+s, rs and s as roots and thereby obtain bounds on the minimum degrees of such polynomials." @default.
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- W2321007863 date "2004-09-01" @default.
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- W2321007863 title "The Arithmetic of Algebraic Numbers: An Elementary Approach" @default.
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