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189-235A: Basic Algebra I

Assignment 5

Due: Wednesday, October 8.

1. Page 162, 3.33.

2-5. Page 171, 3.34, 3.40, 3.41, 3.43.

6-7. Page 178, 3.45, 3.46.

8. Let F be a field having infinitely many different elements.

a) If f and g are two polynomials with coefficients in F, show that if f(x)=g(x) for all x in F, then f=g (as polynomials).

b) Show that this theorem continues to hold if F is only assumed to be a domain (having infinitely many elements) and not necessarily a field.

c) Show (by giving an example) that the conclusion of a) can fail if F is a field with finitely many elements.