Question: Problem 1. Let Q[i] denote the set of complex numbers of the form q+ri where q, r e Q. Show that for any a Q[i]

Problem 1. Let Q[i] denote the set of complex numbers of

Problem 1. Let Q[i] denote the set of complex numbers of the form q+ri where q, r e Q. Show that for any a Q[i] and b E Z[i], a48 + (7 + 14i)a # 216 + 2. (Hint: you may show this by completing the following sketch: Fix b E Z[i], and show that the polynomial 2.48 + (7 + 14i)x 216 2 has no roots in Z[i] as follows: - 248 + (7 + 14i)x 214 2 = 248 2 mod 7. For any a E Z[i], either a is divisible by 7 or a48 =1 mod 7. Be careful: what does "mod mean in this context? Conclude that the polynomial 2:48 + (7 + 14i)x 216 2 has no roots in Q[i]. Alternative solutions are welcome.) Problem 1. Let Q[i] denote the set of complex numbers of the form q+ri where q, r e Q. Show that for any a Q[i] and b E Z[i], a48 + (7 + 14i)a # 216 + 2. (Hint: you may show this by completing the following sketch: Fix b E Z[i], and show that the polynomial 2.48 + (7 + 14i)x 216 2 has no roots in Z[i] as follows: - 248 + (7 + 14i)x 214 2 = 248 2 mod 7. For any a E Z[i], either a is divisible by 7 or a48 =1 mod 7. Be careful: what does "mod mean in this context? Conclude that the polynomial 2:48 + (7 + 14i)x 216 2 has no roots in Q[i]. Alternative solutions are welcome.)

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Accounting Questions!

On September 1, 201X, Dan Potter opened an auto repair shop. Here is his chart of accounts: Assets Liabilities Equity 101 Cash 201 Accounts Payable 301 Dan Potter, Capital 102 Accounts Receivable 302...

1. Let C denote the field of complex numbers. As with any field, we can consider vector spaces, linear transformations, and matrices over C rather than over our usual field R. (a) The complex numbers...

This question concerns lexical grammars. (a) Tree Adjoining Grammars contain two types of elementary tree. (i) What are these trees called? [1 mark] (ii) If one were building a grammar for English...

s1 educated (SSE) student for every three public school educated (PSE) students. Reasoning that students are not very dissimilar from threads, he suggests the following entry and exit routines be...

(a) In SystemVerilog, what is the difference between: (i) The ternary operator ? and if...then...else statements? [2 marks] (ii) always_ff and always_comb? [2 marks] (iii) Blocking, non-blocking and...

Describe, in detail, how the heapsort algorithm works. [10 marks] Show that the worst-case cost of heapsort is O(n log n). [6 marks] Would it be possible to implement a variant of heapsort based on a...

dee complete please help Complexity Theory (a) Defifine the set of Boolean expressions 2CNF and the language 2SAT over them. (b) For a Boolean expression in 2CNF, let G() be the directed graph with...

my economics assign Consider the following economy. There are two periods, 0 and 1, and a single consumption good. The economy is populated by two types of agents: nanciers (F) and entrepreneurs (E)....

Define the notions of Turing-recognisable language and undecidable language. b) Is L Turing- recognisable? Justify your answer with an informal argument. c) Justify with a formal proof your answer to...

(a) How does the use of condition codes complicate the implementation of a superscalar processor that supports out-of-order execution? [4 marks] (b) A branch predictor with a high prediction accuracy...

please solve this problem.. Needed all subparts will be really appreciated.. 5. Let's call a 2 X 2 matrix complicated if it has the form [ a b] where a and b are real numbers. bo (a) [3 pts] Show...

1. Describe the possible echelon forms of a nonzero 2 x 2 matrix. Use the symbols , *, and 0, as in the first part of Example 1. 2. Repeat Exercise 5 for a nonzero 3 x 2 matrix.

Government Hospital Financial Reporting Classifications Using the preceding abbreviations, indicate how each of the following items should be reported in these two hospital financial statements. If...

Evaluate the iterated integral. 0 9 0 4 0 1 6 - x 2 2 x z + 1 d y d x d z

An advertisement seeking volunteers for a clinical research draws 11 respondents. Of these respondents, 5 are below age 30 and 6 are over 30. The researcher will randomly select 4 persons to assign...

An IRS agent receives a batch of 18 tax returns that were flagged by computer for possible tax evasions. Suppose, unknown to the agent, 7 of these returns have illegal deductions and the other 11 are...

Balthus Corp. reports the following components of stockholders' equity on December 31, 2015: Common stock-$1 par value, 320,000 shares authorized, 200,000 shares issued and outstanding . . . . . . ....

The equity sections from Hovo Corporation's 2015 and 2016 balance sheets follow. Stockholders' Equity (December 31, 2015) Common stock-$20 par value, 30,000 shares authorized, 17,000 shares issued...

Consider two mutually exclusive R&D projects that ADM is considering. Assume the discount rate for ADM is 15 percent. Project A: Server CPU .13 micron processing project By shrinking the die size to...

Realiza un cuadro comparativo donde analices y expliques: Las diferencias y semejanzas entre los diferentes tipos de auditora incluyendo la auditoria adminstrativa

=+4. Look at Marks & Spencers Plan A Web site (http://plana.marksandspencer.com/). What are your impressions? Do

=+3. Go to the Web site for the movie documentary, The Corporation (http://www.thecorporation.com/). Have a look at the trailer posted on the site.195 Is this kind of commentary helpful to those...

=+4. Think of an example of a corporation that you feel has been hypocritical in terms of its ethical or CSR stance. What is your reaction to the companys actions? Would you consider boycotting the...