1.39. Definition. Field Axioms. A set S with operations + and . and distinguished elements 0 and 1 with 0 * 1 is a field if the following properties hold for all x, y, z E S. AO: x+ y ES MO: X . YES Closure Al: (xty)+z = x+(y+z) M1: (x . y) . z = x . (y . z) Associativety A2: xty = y+ x M2: x . y =y . x Commutativety A3: x + 0 = x M3: x . 1= x Identity A4: given x, there is a wes M4: for x * 0, there is a wes Inverse such that x + w = 0 such that x . w = 1 DL: x . (y + z) = x . y + x . z Distributive Law The operations + and . are called addition and multiplication. The elements 0 and 1 are the additive identity element and the multiplicative identity element. It follows from these axioms that the additive inverse and multiplica- tive inverse (of a nonzero x) are unique. The additive inverse of x is the negative of x, written as -x. To define subtraction of y from x, we let x-y=x+(-y). The multiplicative inverse of x is the reciprocal of x, written as x . The element 0 has no reciprocal. To define division of x by y when y * 0, we let x/y = x . (y !). We write x . y as xy and x . x as x2. We use parentheses where helpful to clarify the order of operations. 1.40. Definition. Order Axioms. A positive set in a field F is a set P C F such that for x, y e F, P1: x, y E P implies x + y e P Closure under Addition P2: x, y E P implies xy e P Closure under Multiplication P3: x E F implies exactly one of Trichotomy x =0, xEP, -x EP An ordered field is a field with a positive set P. In an ordered field, we define x have analogous definitions in terms of P. Note that P = (x e F: x > 0). Another phrasing of trichotomy is that each ordered pair (x, y) satisfies exactly one of x y.Order and justify these steps to prove the well-loved property that "zero times anything equals zero" for a generic field F (Proposition 1.43b in D'Angelo & West). Assume that r E F and 0 is the additive identity. Use the properties on page 16. Claim Justification 0 . r +0 .r = (0+0) .I 0. r+0 . r =0 . I (0 . r +0 . x) + -0 .1=0 . r+-0 .I 0 . x+(0 . x+-0 .1)=0 . r+-0.1 0 . 2 +0=0 0 . x=0