Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

376 colin mclartyAnd the equation has solutions of other forms. For example, in numbers of theform a + b √ 2 with ordinary integers a, b there is a solution X = 0, Y = √ 2since( √ 2) 2 = 3 · 0 + 214.1.4 Arithmetic via morphismsThis reasoning can be organized in the ring Z/(3), thequotient ring of Z by 3,which is the set {0, 1, 2} with addition and multiplication by casting out 3s:+ 0 1 20 0 1 21 1 2 02 2 0 1· 0 1 20 0 0 01 0 1 22 0 2 1Construing Equation (14.1) inZ/(3) makes 3 = 0andsogivesY 2 = 2. Themultiplication table shows 0,1 are the only squares in Z/(3).SoEquation(14.1)has no solutions in Z/(3). Next, consider the function r : Z → Z/(3) takingeach integer to its remainder on division by 3. Sor(0) = r(3) = 0 r(2) = r(8) = 2 and so onBy Theorem 5, this preserves addition and multiplication. So it would take anyinteger solution to any polynomial equation over to a solution for that sameequation in Z(3). Since Equation (14.1) has no solutions in Z(3), it cannothave any in Z either.The watershed in ‘modern algebra’ came when Noether reversed theorder of argument. Instead of beginning with arithmetic she would deducethe arithmetic from purely structural descriptions of structures like Z/(3)(Noether, 1926).InthecaseofZ/(3) this meant looking at arbitrary ringsand ring morphisms: A ring R is any set with selected elements 0,1 andoperations of addition, subtraction, and multiplication satisfying the familiarassociative, distributive, and commutative laws.⁷ A ring morphism f : R → R ′ isa function preserving 0,1 addition and multiplication. Noether would rely onthe following:Fact on Z/(3) The ring morphism r : Z → Z has r(3) = 0 and: For any ring R andring morphism f : Z → Rwithf(3) = 0 there is a unique morphism u : Z/(3) → Rwith ur = f .⁷ See Mac Lane (1986, pp. 39, 98) or any algebra text. All rings in this paper are commutative.