## Chapter 3 Exercises## 3.1 Polynomials- 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? - Prove that addition and multiplication of polynomials is commutative.
- Prove that addition and multiplication of polynomials is associative.
- 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?
- 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.
- Give a complete proof of Proposition 2.
- State and prove an assertion like Proposition 2 which applies to rational functions.
- 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.
- Adapt Euclid's proof of the theorem that there are infinitely many primes to polynomials over a field F.
- Can you adapt the proof of Euclid's theorem using Fermat numbers to polynomials over a finite field F?
## 3.2 Rational Roots- 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- Prove Corollary 2.
- Complete the proof of Proposition 4.
- Prove Corollary 3.
- Prove Corollary 4.
## 3.4 Cubic Equations## 3.5 Quartic EquationsAll contents © copyright 2001 K. K. Kubota. All rights reserved |