9.2 ORDERED FIELDS 303(b) (i) Construct addition and multiplication tables for the field Z, of integers
modulo 7.
(ii) Find all solutions in Z, of the equation x3 + x2 - 2x = 0.- (a) Prove that the zero and unity elements in a field equal their own additive and
multiplicative inverses, respectively. That is, prove that - 0 = 0 and 1 - ' = 1.
(b) Prove or disprove: In any field F, if x E F and x = x- ', then x = 1.
(c) Prove or disprove: In any field F, if x E F and x = - x, then x = 0.
(d) Let m be a positive integer. Give an example of a field F and an element
x E F such that mx = 0 (where we define mx recursively by 1 x = x and mx =
(m - 1)x + x if m > 1; recall Definition 4) but x # 0. - (a) Prove that if F is a field and if x and y are nonzero elements of F (so that
xy # 0; recall Theorem 4), then (x y) - ' = x - ' y - '.
(b) Prove that if x is a nonzero element of a field F, then (x-')-' = x. - (a) Prove that if F is a field and x E F satisfies x2 = x, then either x = 0 or
x= 1.
(b) Prove that if F is a field and x E F satisfies x2 = 1, then x = f 1. Does this
result hold true in the structure Z8? - (a) Prove that if x, y, z E F (F a field) with z # 0, then (xlz) + (ylz) = (x + y)/z.
(b) Show that if a, b, c, and d are elements of F (F a field), then (a + b)(c + d) =
ac + ad + bc + bd.
(c) Prove parts (a) through (f) and (i) of Theorem 5.
(d) Prove Theorem 6. - (a) Prove, in a field F, if a, b, c, x E F and a # 0, then ax + b = c if and only
if x = (c - b)a- l. Hence a linear equation in one variable with coefficients in a
field F has a unique solution in F.
(b) Prove that if F is a field and a, b E F, then a2 = b2 if and only if a = b or
a= -b.
(c) Give an example in the structure Z8 of a linear equation ax + b = c that has:
(i) no solution x in Z, (ii) more than one solution in Z,
(d) Find an example of elements a, b E Z, such that a2 = b2, but a # b and a # - b. - Prove or disprove, in an arbitrary field F:
*(a) If a E F, then there corresponds x E F such that x2 = a (such an element x
would be called a square root of the field element a).
(b) If x, y E F and x2 + y2 = 0, then x = y = 0. - Verify field axibms 7, 10, and 11 for the field ~[a].
9.2 ORDERED FIELDS
We saw in Article 9.1 that R is a field and is thereby distinguishable from
its subsets N and Z. But the theory of general fields fails to provide a means
of differentiating between R and either Q or C. Like the reals, the rational
and complex number systems are themselves fields.