Essays in Constructive MathematicsHe [Kronecker] was, in fact, attempting to describe and to initiate a new branch of mathematics, which would contain both number theory and alge braic geometry as special cases.—Andre Weil [62] This book is about mathematics, not the history or philosophy of mathemat ics. Still, history and philosophy were prominent among my motives for writing it, and historical and philosophical issues will be major factors in determining whether it wins acceptance. Most mathematicians prefer constructive methods. Given two proofs of the same statement, one constructive and the other not, most will prefer the constructive proof. The real philosophical disagreement over the role of con structions in mathematics is between those—the majority—who believe that to exclude from mathematics all statements that cannot be proved construc tively would omit far too much, and those of us who believe, on the contrary, that the most interesting parts of mathematics can be dealt with construc tively, and that the greater rigor and precision of mathematics done in that way adds immensely to its value. |
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algebraic curve algebraic field algebraic number algebraic number field algebraic quantities algebraic variations Ax² canonical form characteristic polynomial comparison algorithm complex numbers computations construction contains curve defined described determine diagonal divides divisible element entries equation equivalent Essay expansion exponents factorization of f(x field of g(y field of rational follows formula g mod g₁ Galois field Galois group Gauss's genus given greatest common divisor holomorphic differentials hypernumbers implies indeterminates integer coefficients integral basis integral domain irreducible factors irreducible polynomial Kronecker Kronecker's lemma linear factors mathematics matrix mod F mod f(x mod g(y monic polynomial multiplication Newton's polygon poles polynomial f(x polynomial of degree polynomial with coefficients polynomial with integer proof Proposition proved rational coefficients rational functions rational numbers real number relatively prime root field sequence shown splitting field truncated solution