Reading: Text: Sections 1.4-6.
Problems: Section 1.4: 7, 16.
Section 1.5: 2, 4(b), 10, 12.
Section 1.6: 2(d), 5 (see the definitions
of the non-Euclidean norms in Exercise 4).
Problem C: Prove that there is no way to introduce an order on the field C of complex numbers so that it becomes an ordered field. Hint: Shall we have i > 0 or < 0?
Problem D: Show that any automorphism of the field R of real numbers is trivial, i.e., the identity. An automorphism is a bijection which respects the addition and multiplication. Hint: First, show that the rationals must be fixed by an automorphism. Then show that an automorphism must preserve the order of the real numbers. (Surprise: algebra enforces analysis!)
Last modified: (2015-09-18 18:12:30 CDT)