S OCIAL E COLOGY W EEK 5 WORKSHEET
M ATH 120B (45042) W INTER 2017
1. Apply the division algorithm to f (x) = x4 + x3 + x2 + x + 1 and g(x) = 3x2
polynomial ring Z5 [x].
1 in the
2. Find an example to show that the division algorithm does not hold in S[x] if we require only that
S be an integral domain.
3. We say two functions f (x), g(x) are equal if f (a) = g(a) for every input a.
a. Consider the functions corresponding to polynomials f (x), g(x) 2 R[x]. True/False: If
the functions f (x), g(x) are equal, then the polynomials f (x), g(x) are also equal. (Hint.
Subtract one polynomial from the other. You should use the result that if the coefficients are
in a field, then a degree n polynomial has at most n zeros.)
b. Consider the functions corresponding to polynomials f (x), g(x) 2 F [x], where F is some
field. True/False: If the functions f (x), g(x) are equal, then the polynomials f (x), g(x) are
also equal. (What part of your proof fails?)
c. Consider the functions corresponding to polynomials f (x), g(x) 2 R[x], where R is some
commutative ring with unity. True/False: If the functions f (x), g(x) are equal, then the
polynomials f (x), g(x) are also equal.
4. Recall that ev↵ denotes the evaluation homomorphism obtained by evaluating at ↵.
a. Find a degree 3 polynomial in the kernel of evi : R[x] ! C, or prove that none exists.
b. Find a degree 1 polynomial in the kernel of evp2 : R[x] ! R, or prove that none exists.
c. What is the kernel of ev⇡ : Q[x] ! R?
d. Let ↵,
e. Let ↵,
2 R. Prove that if ev↵ , ev : R[x] ! R are equal as functions, then ↵ = .
2 C. If ev↵ , ev : R[x] ! C are equal as functions, is it necessarily true that ↵ = ?
f. Is there a field F and an element ↵ 2 F such that the kernel of ev↵ is the ring F [x]?
g. Let ↵ 2 C and assume ev↵ : Q[x] ! C has image contained in R. Prove that ↵ 2 R.
5. Prove that the evaluation homomorphism ev↵ : R[x] ! C is never injective for any ↵ 2 C.
6. a. Prove or disprove: The ring R[x] has infinitely many units.
b. Prove or disprove: The ring Z[x] has infinitely many units.
c. Prove or disprove: The ring Z5 [x] has infinitely many units.
d. Prove or disprove: The ring Z6 [x] has infinitely many units.