Question

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, 2 is prime in J. Show −2 is also prime.

Answer #1

Then

As if neither of a and b are units we must have

If as is not possible which makes which is also impossible

So meaning at one of them is a unit

Similarly, if and similar logic tells us that

meaning at one of them is a unit

Thus, 2 and -2 are primes in J

Solve linear systems over complex numbers
(1-i)x + (2+i)y = 4i - 2
(-2+2i)x + (3i)y = -7i -7

Prove or disprove (a) Z[x]/(x^2 + 1), (b) Z[x]/(x^2 - 1) is an
Integral domain.
By showing (a) x^2+1 is a prime ideal or showing x^2 + 1 is not
prime ideal.
By showing (b) x^2-1 is a prime ideal or showing x^2 - 1 is not
prime ideal.
(Hint: R/I is an integral domain if and only if I is a prime
ideal.)

Using field and order axioms prove the following theorems:
(i) Let x, y, and z be elements of R, the
a. If 0 < x, and y < z, then xy < xz
b. If x < 0 and y < z, then xz < xy
(ii) If x, y are elements of R and 0 < x < y, then 0 <
y ^ -1 < x ^ -1
(iii) If x,y are elements of R and x <...

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.

Let D be the solid region defined by D = {(x, y, z) ∈ R3; y^2 +
z^2 + x^2 <= 1},
and V be the vector field in R3 defined by: V(x, y, z) = (y^2z +
2z^2y)i + (x^3 − 5^z)j + (z^3 + z) k.
1. Find I = (Triple integral) (3z^2 + 1)dxdydz.
2. Calculate double integral V · ndS, where n is pointing
outward the border surface of V .

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
x = {x} and y ={y} represent bounded sequences of real numbers, z =
x + y, prove the following: supX + supY = supZ where sup represents
the supremum of each sequence.

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A 4
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

Let X, Y ⊂ Z and x, y ∈ Z
Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

3. Let N denote the nonnegative integers, and Z denote the
integers. Define the function g : N→Z defined by g(k) = k/2 for
even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a
bijection.
(a) Prove that g is a function.
(b) Prove that g is an injection
. (c) Prove that g is a surjection.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 15 minutes ago

asked 16 minutes ago

asked 19 minutes ago

asked 26 minutes ago

asked 41 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago