1.8. COMPLEX NUMBERS 19 Exercise 1.7.7. (i) Find Aut (Fp) where p is a prime and Zp is the finite field with p elements. (ii) Let F = {a + b √ 2 | a, b ∈ Q}. Show that F is a field and find Aut (F ). This is the beginning of the subject called Galois theory, in which one of the goals is to determine Aut (F ) when F is a so-called “algebraic extension” of Q. 1.8. Complex Numbers To start this section, we give a somewhat inexact definition of complex numbers. This is often used as a definition of the complex numbers, but it does contain some ambiguity, which we will rectify immediately. Definition 1.8.1 (Rural definition). The set of complex numbers, C, is the collection of expressions of the form z = a + bi where a, b ∈ R and i is a symbol which satisfies i2 = −1. If z = a + bi and w = c + di are in C, then we define z + w = (a + c) + (b + d)i and zw = (ac − bd) + (bc + ad)i. Actually, one can go a long way with this definition if the symbol i with the property that i2 = −1 does not cause insomnia. To be more precise, we consider the Cartesian product R×R with addition defined by (a, b)+(c, d) = (a + c, b + d) and multiplication defined by (a, b)(c, d) = (ac − bd, bc + ad). Exercise 1.8.2. Show that R × R with addition and multiplication as defined above is a field with (0, 0) as the additive identity, (1, 0) as the mul- tiplicative identity, −(a, b) = (−a, −b), and (a, b)−1 = (a/(a2 +b2), −b/(a2 + b2)) if (a, b) = (0, 0). So R × R with these operations forms a field which we denote by C and call the field of complex numbers. Note that R is isomorphic to the subfield of C given by {(a, 0) | a ∈ R}. If we set i = (0, 1), then i2 = (−1, 0). Finally, to fix things up really nice, we write (a, b) = (a, 0)+(b, 0)(0, 1), or, returning to our original rural definition, (a, b) = a + bi. The first observation to make is that C cannot be made into an ordered field. That is, it cannot satisfy the order axioms given in Appendix A. This is immediate because in any ordered field, if a = 0, then a2 0. This would imply that i2 = −1 0, but 12 = 1 0, and this is a contradiction. Definition 1.8.3. If z = a + bi, we call a the real part of z and b the imaginary part of z. We write a = Re z and b = Im z. The complex number z is called pure imaginary if a = Re z = 0. Definition 1.8.4. If z = a + bi, the complex conjugate of z, denoted ¯, is ¯ = a − bi. The absolute value of z is |z| = (z¯) 1 2 = (a2 + b2) 1 2 , where, of course, we mean the nonnegative square root in R.

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