Exam

PMATH 334 Final Exam August 7, 2007 Time allowed: 150 minutes Instructions 1. Attempt problems 1–5, and either problem 6 or problem 7. If you do both problems 6 and 7, only the better solution will be counted. 2. Problems 1 and 2 are worth 5 points each; problems 3 to 5 are worth 10 points each; problems 6 and 7 are worth 20 points each. For multi-part problems, the value of each part is shown in [square brackets] after the statement of that part. Except as indicated above, the problems are in no particular order. 3. Be sure to show work, for otherwise it is impossible for me to give partial credit. 4. Calculators are not permitted. 5. You may detach the last two pages of this booklet and use them for rough work. If you would like anything on these sheets to be considered for credit, please enclose the paper in your test and indicate clearly on the test which problem the material on the paper refers to. I’d prefer that you didn’t do this, though. 6. This test should have 11 pages, including the cover page and this page. If it does not, please request a replacement paper. 7. Good luck!

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√ √ √ 1. Let R be the ring Z[ 3 2] = {a + b 3 2 + c 3 4 | a, b, c ∈ Z}. Prove that S is a subring of R, where √ √ 3 3 S = {a + b 2 + c 4 | a, b, c ∈ Z, 2|b}.

(You may assume and need not prove that R is a ring.)

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[5]

2. Factor the polynomial x5 + x4 + x3 + 2x2 + 1 ∈ Z3 [x] into irreducible factors. [5]

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√ 3. Let R = Z[ 5] and let I = (2). (i) List the elements of R/I. [3] (ii) Make a multiplication table for R/I. [4] (iii) Find an ideal of R that contains I and is not equal to either R or I. [3]

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4. Let p = x3 − 4x + 2 ∈ Q[x]. (i) Prove that p is irreducible. [3] (ii) Let s be the class [x2 + 2x − 1] in Q[x]/(p(x)). Find a minimal polynomial of s. [6] (iii) Describe all minimal polynomials of s. [1]

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5. Let C be the code of length 6 over Z3 with basis (102211), (220120), (011111). (i) Find a check matrix for this code. [2] Also, state its information rate. [1] (ii) State and verify a condition for this code to correct one error. [3] (iii) Assuming at most one error, decode the following words: (001221), (201201). [2 each]

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√ 6. Let R = Z[ 7]. The √ goal of this problem is to prove that R is a Euclidean domain√for the usual norm function N (x + y 7) = |x2 − 7y 2 |. Recall that N (rs) = N (r)N (s) for all r, s ∈ Q( 7). √ (i) Let a, b 6= 0 ∈ R. Prove that there exist elements q and r of R such that a = bq+r and r = b(s+t 7) with |s|, |t| ≤√1/2. [4] (ii) Let l = s + t 7 with |s|, |t| ≤ 1/2 as above. Show that if 1 < 7t2 < 5/4 then either N (l + 1) < 1 or N (l − 1) < 1. [4] (iii) With l, s, t as above, show that if 5/4 < 7t2 then either N (l + 1) < 1 or N (l − 1) < 1. [4] (iv) Combining (i), (ii), and (iii) above, prove that R is a Euclidean domain. (Be sure to explain why 7t2 6= 1, 5/4.) [5] √ (v) Using the Euclidean algorithm, determine gcd(1 + 7, 8). [3]

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√ √ 7. In this problem, let R = Z[ 6] and let I be the ideal (3, 6). (i) Prove that I 2 = (3). [8] (ii) Find (with proof) a ∈ R such that I = (a). [6] (iii) Using your result from (ii), find a generator for I 2 that is neither 3 nor −3. [3] (iv) Using your results from (i) and (iii), find a unit in R that is neither 1 nor −1. [3]

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