Chapter 3 Exercises

3.1 Polynomials

1. Let F be a field. The product of two polynomials and is defined to be . The number of terms in the inner sum depends on the parameter k. For example, the constant term k = 0 and the highest degree term both have only one inner summand. Show that the number N(k) of terms in the inner sum for the term of degree m + n - k is What is the largest number of terms in any of the inner sums?

2. Prove that addition and multiplication of polynomials is commutative.
3. Prove that addition and multiplication of polynomials is associative.
4. What are the additive and multiplicative identities in the field of rational functions with coefficients in a field F? If p(x)/q(x) is a rational function, what is its multiplicative inverse? What is its additive inverse?
5. The statement of Proposition 2 limits its statement about the uniqueness of factorization to non-zero monic polynomials. Extend it so that it is a statement about the uniqueness of the factorization of arbitrary polynomials.
6. Give a complete proof of Proposition 2.
7. State and prove an assertion like Proposition 2 which applies to rational functions.
8. For the integers, one defined least common divisor and provided the Euclidean algorithm as a means for calculating the least common divisor of two non-zero integers. Make similar definitions and give an analogue of the Euclidean algorithm for polynomials over a field F. Show that the algorithm does indeed calculate the least common divisor of two polynomials.
9. Adapt Euclid's proof of the theorem that there are infinitely many primes to polynomials over a field F.
10. Can you adapt the proof of Euclid's theorem using Fermat numbers to polynomials over a finite field F?

3.2 Rational Roots

1. State and prove a result analogous to Proposition 3 where the coefficients of the equation are polynomials (in some variable y) over a field F.

3.3 The Fundamental Theorem of Algebra

1. Prove Corollary 2.
2. Complete the proof of Proposition 4.
3. Prove Corollary 3.
4. Prove Corollary 4.

3.5 Quartic Equations

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Revised: July 10, 2001