Let () be a nonzero principal ideal in Z i a Show that Z i () is a finite ring b Show that if is an irreducible of Z i , then Z i () is a field c Referring to part (b), find the order and characteristic of each of the following fields i Z i (3) ii Z i (1 i) iii Z i (1 2i) ...

a Let be a coset of Zi By the division algorithm where either 0 or N N Then Now so hi Thus every cos ...

Let () be a nonzero principal ideal in Z[i]. a. Show that Z[i]/() is a finite ring.

Question:

Let (α) be a nonzero principal ideal in Z[i].

a. Show that Z[i]/(α) is a finite ring.

b. Show that if π is an irreducible of Z[i], then Z[i]/(π) is a field.

c. Referring to part (b), find the order and characteristic of each of the following fields.

i. Z[i]/(3)

ii. Z[i]/(1 + i)

iii. Z[i]/(1 + 2i)

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

A First Course In Abstract Algebra

ISBN: 9780201763904

7th Edition

Authors: John Fraleigh

See More Books

Question Posted: Jan 11, 2023 04:42 AM

See More Questions

Show that if E is a finite extension of a field F and [E : F] is a prime number, then E is a simple extension of F and, indeed, E = F() for every a E not in F.
Show that if G is a finite group with identity e and with an even number of elements, then there is a e in G such that a * a = e.
Let z be a nonzero complex number and n a negative integer (n = 1,2, . . .). Also, write z = reiÎ¸ and m = n = 1, 2, . . . . Using the expressions verify that (zm)1 = (z1)m and hence that...
Define the necessary and sufficient condition for two transactions to be serialisable. Give an example of a non-serialisable execution of a pair of transactions. [3 marks] (c) Define the necessary...
Briefly explain how increasing sample size influences each of the following. Assume that all other factors are held constant. a. The size of the z-score in a hypothesis test. b. The size of Cohen's...
Use the results of Exercises 7475 to describe and graph the following circles. r 2 - 2r(2 cos + 3 sin ) = 3
A parallel-plate capacitor connected to a battery maintaining a potential difference $V$ across the capacitor initially stores electric potential energy $U_{1}^{E}$. If the plate area is doubled...
The Arkade Company is considering switching to a 15 percent allowance based on time worked. Calculate the new standard time for each work element and for the totaljob. Mean Observed Time minutes 1.20...
A student addresses a bully mocking another student, " what you're doing isn't even clever or funny, everyone has a right ti express who they are.. you know like we say every morning.. liberty and...
Let D be a Euclidean domain and let v be a Euclidean norm on D. Show that for nonzero a, b D, one has v(a) < v(ab) if and only if b is not a unit of D.
Express the given polynomial as the product of its content with a primitive polynomial in the indicated UFD. 18x 2 - 12x + 48 in Q[x]
Convert a temperature of 80F to: a. C b. R c. K
How do executive-level decisions affect organizational change? How should an executive cast vision for a major change? For example, when an executive wants to execute change in the organization, what...
Marisa wants to buy a home in Atlanta with a 30-year mortgage that has an annual interest rate of 4.9%. The house she wants is $250,000, and she will make a $55,000 down payment and borrow the...
A bond is currently trading at $952.31. The bond has a face value of $1,000, a coupon rate of 8% with payments made semi-annually, and mature in 20 years. What is the yield to maturity? ( Company A's...
A force is given as F=-yzi -xzj-xyk (a) Show that this force is conservative. (b) Calculate the corresponding potential energy V as a function of x, y, and z. (c) Show that the mechanical energy E...
A company sells a piece of real estate it owns and receives $20.0 million in cash which includes a reported gain on sale of $1.5 million on its income statement. How would the statement of cash flows...
How would an advertising plan for an online-only cosmetics retailer differ from that of a bricks-and-mortar cosmetics store chain?
How do the principles of (a) Physical controls and (b) Documentation controls apply to cash disbursements?
Reconsider the previous exercises about reaction times and time of day. a. Find an approximate 95% confidence interval for the mean difference in the reaction times (morning night) for the population...
Reconsider the previous exercises about reaction times and time of day. The statistic used was the mean of the difference in reaction times (morning night). Explain how the following things would...
Students wanted to see whether exercising (in particular doing jumping jacks) would help or hinder peoples ability to memorize a list of 10 words that were read to them. This would be compared to the...
Why is a visual analysis important in communicating findings in a health care research paper or proposal? Give an example of a visual analysis that would be appropriate for your paper and...
2) A firm is evaluating a project with the following cash flows: Year 0: - 28000 Year 1: 12000 Year 2: 15000 Year 3: 11,000 a) If the required return is 14%, what is the NPV? b) And what is the IRR...
2. Refer to the figure below for all remaining problems. The footing is square with a width of 5' and the loading is concentric. Laboratory strength and consolidation tests performed at various...

Let () be a nonzero principal ideal in Z[i]. a. Show that Z[i]/() is a finite ring.

Question:

Step by Step Answer:

a Let be a coset of Zi By the division algorithm where either 0 or N N Then Now so hi Thus every cos...View the full answer

A First Course In Abstract Algebra

Students also viewed these Mathematics questions