Question

1. Let Z[i] denote the set of all ‘complex numbers with integer coefficients’:the set of all a + bi such that a and b are integers. We say that z is composite if there exist two complex integers v and w such that z=vw and |v|>1 and |w|>1. Then z is prime if it is not composite

A) Prove that every complex integer z, |z| > 1, can be expressed as a product of prime complex integers.

Answer #1

Definition:In the complex numbers, let J denote the set, {x+y√3i
:x and y are in Z}. J is an integral domain containing Z. If a is
in J, then N(a) is a non-negative member of Z. If a
and b are in J and a|b in J, then N(a)|N(b) in Z. The units of J
are 1, -1
Question:If a and b are in J and ab = 2, then prove one of a and
b is a unit. Thus,...

Complex Variables:
(a) Describe all complex numbers 'z' such that e^z = 1.
(b) Let 'w' be a complex number. Let 'a' be a complex number
such that e^a = w. Describe all complex numbers 'z' such that e^z =
w.

Let Z[x] be the ring of polynomials with integer coefficients.
Find U(Z[x]), the set of all units of Z[x].

Due October 25. Let Z[i] denote the Gaussian integers, with norm
N(a + bi) = a 2 + b 2 . Recall that ±1, ±i are the only units i
Z[i]. (i) Use the norm N to show that 1 + i is irreducible in Z[i].
(ii) Write 2 as a product of distinct irreducible elements in
Z[i].

Find the set of complex numbers z satisfying the two conditions:
Re((z+1)^2)=0 and Im((z−1)^2)=2. Here Re(a + bi) = a and Im(a + bi)
= b if both a, b ∈ R. Then find the cardinality of the set.

Let f ∈ Z[x] be a nonconstant polynomial. Prove that the set S =
{p prime: there exist infinitely many positive integers n such that
p | f(n)} is infinite.

Problem 2: (i) Let a be an integer. Prove that 2|a if and only
if 2|a3.
(ii) Prove that 3√2 (cube root) is irrational.
Problem 3: Let p and q be prime numbers.
(i) Prove by contradiction that if p+q is prime, then p = 2 or q
= 2
(ii) Prove using the method of subsection 2.2.3 in our book that
if
p+q is prime, then p = 2 or q = 2
Proposition 2.2.3. For all n ∈...

4. (30) Let C be the ring of complex numbers,and letf:C→C be the
map defined by
f(z) = z^3.
(i) Prove that f is not a homomorphism of rings, by finding an
explicit counterex-
ample.
(ii) Prove that f is not injective.
(iii) Prove that the principal ideal I = 〈x^2 + x + 1〉 is not a
prime ideal of C[x].
(iv) Determine whether or not the ring C[x]/I is a field.

Prove: If f(x) = anx^n + an−1x^n−1 + ··· + a1x + a0 has integer
coefficients with an ? 0 ? a0 and there are relatively prime
integers p, q ∈ Z with f ? p ? = 0, then p | a0 and q | an . [Hint:
Clear denominators.]

1. Let W be the set of all [x y z}^t in R^3 such that xyz = 0.
Is W a subspace of R^3?
2. Let C^0 (R) denote the space of all continuous real-valued
functions f(x) of x in R. Let W be the set of all continuous
functions f(x) such that f(1) = 0. Is W a subspace of C^0(R)?

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