Question

Definition:In the complex numbers, let J denote the set, {x+y√3i :x and y are in Z}....

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.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

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

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
1. Let Z[i] denote the set of all ‘complex numbers with integer coefficients’:the set of all...
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.
Solve linear systems over complex numbers (1-i)x + (2+i)y = 4i - 2 (-2+2i)x + (3i)y...
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...
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...
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...
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...
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'...
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...
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}....
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}....
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 )
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT