## Transcribed Text

1. Using Euclid’s algorithm for polynomials, find the gcd of the polynomials
p(x) = x
9 − 2x
8 + x
7 + x
5 − 4x
4 + 5x
3 + 15x
2 − 34x + 17
q(x) = x
4 − 2x
2 + 1
2. (a) Consider Z[i] = {a + ib|a, b ∈ Z}, a subring of C. By considering the numbers of C as points in the plane and looking at the shape made by the points
which lie in Z[i], prove the following : for every complex number z, we can
find a q ∈ Z[i] such that |z − q| ≤ 1, where |a + ib| =
√
a
2 + b
2
.
(b) For a, b ∈ Z[i], apply part (a) of this exercise to z = a/b to conclude that
Z[i] is an euclidean domain.
3. (a) Let R be a domain, let f be a size function on R and let u ∈ R. Prove that
u is a unit if and only if f(u) = 1. (Hint : Find f(1) first)
(b) Find all units of Z[i] and of F[x], where F is a field.
4. Prove that 4 + i is an irreducible element of Z[i]. (Hint : Use the size function on
Z[i] that we discussed in the class)
5. (a) Prove that x
2+1 is an irreducible element of Q[x]. Is it an irreducible element
of C[x]?
(b) Find a factorization of x
4 + x
2 + 1 into irreducible elements in Q[x]. (Hint :
x
4 + x
2 + 1 = (x
4 + 2x
2 + 1) − x
2
)
6. (a) Let F be a field. Let a, b ∈ F such that a 6= 0. Then prove that ax + b is an
irreducible element of F[x].
(b) Let F be a field and R = F[x] is the ring of polynomials with coefficients
in F. Suppose that p(x) ∈ F[x] is a polynomial. Let a ∈ F. Then we can
define polynomial l(x) = x − a. Show that l|p if and only if p(a) = 0.
(c) Use part(b) to prove that x
3 − 2 is irreducible in Q[x].
7. Suppose that R is an integral domain and that the number of elements of R is
finite. Let a ∈ R be a fixed non-zero element. Show that the multiplication-by-a
map, g : R → R, g(x) = ax must be injective. Using the fact that the cardinality
of R is finite and g is injective, prove that g is surjective. Using this, show that
R is a field.

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