M300.3 Final Exam

发布时间：2023-12-18

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Final Exam

M300.3

Rules: you can discuss with your fellow students, look up notes, chapters in books, google things if you need to. Everything we have proven in class does not need to be reproved, you can just cite it (The- orem/Proposition/Lemma so and so/about so and so from class). If you use any other results not covered in class in your answers, you must provide a complete proof of those results before you apply them to your answer.

You have to use the notation and concepts from this class only. If you copy/paste solutions you ind on the web, chances are that these use diferent notations etc. And it is very easy for me to see if you just copy a proof from somewhere without explaining it in your own language, in your own understanding.

Try as best as you can to write coherently, explain your thought process and structure of proof as clearly as you can.

Problem 1. Let G be an Abelian group and H G a subgroup. We write the group multiplicatively, that is, the group operation is notated by g1g2 for gk 2 G, the neutral element is notated by 1, and the inverse element to g 2 G by g-1 . Consider the set

G/H = fgH ; g 2 Gg

of left cosets of H.

(i) Show that

g1 H · g2 H := g1g2 H

is well-deined on G/H, that is, if gkH =˜(g)kH , k = 1, 2, then g1g2 H =˜(g)1˜(g)2 H.

(ii) Show that G/His an Abelian group by determining the neutral and inverse elements and checking the group axioms.

(iii) Show that the map π : G ! G/H, given by π(g) = gH, is a surjective group homomorphism.

(iv) Determine ker π .

(v) What is G/H for G = (Z, +) and H = nZ = fnk ; k 2 Zg Z? Note that in this example we write the group additively.

Problem 2. Consider the ield with two elements Z2 = f0, 1g and consider GL2 (Z2 ), the group of invertible 2 X 2 matrices with entries in Z2 . Provide an explicit group isomorphism between GL2 (Z2 ) and the permutation group of S3 .

Hint: First calculate all elements in GL2 (Z2 ). Observe that |GL2 (Z2 )| = |S3 | . Then consider Z2 X Z2 on which GL2 (Z2 ) acts in the usual way:

matrix multiplies vector. How many elements does Z2 X Z2 have? How is this related to permutations of a set of three elements?

Problem 3. Let p = 3, 5, 7, 11. Show that (Zp(根) , ·) is a cyclic group by providing a generator g ∈ Zp(根) so that

{gk ; k = 0, . . . , p — 2} = Zp(根)

Thus (Zp(根) , ·) has to be isomorphic to (Zp-1 , +). Write down an explicit

group isomorphisms for each case.

Problem 4. Let φ: N → N be Euler’s totient function deined by

φ(n) := number of elements in {1,2,. . . ,n} for which gcd(k, n) = 1.

Show:

(i) φ(n) = |Zn(根) | .

(ii) φ(p) = p — 1 for p prime.

(iii) φ(pm ) = pm — pm-1 for p prime and m ∈ N.

Hint: in (iii) list all elements in Zpm and ind all elements [k] for which gcd(k, n) 1 (do this for small m to get an idea how it works).

Problem 5. Let p > 2 be prime and 0 a ∈ Zp.

(i) If the equation x2 = a has a solution in Zp , then = 1 in Zp. Hint: Fermat’s Little Theorem.

(ii) Prove the converse of (i) in the case p 三 3 mod 4. Hint: why

(iii) Show that the equation x2 = a has either no solution or exactly two solutions. Hint: show irst that in any ield ab = 0 implies a = 0 or b = 0. Apply this to show that the equation x2 = y2 in any ield for x 0 has exactly two solutions: x2 = y2 is equivalent to x2 — y2 = 0...

Problem 6. Find all solutions of the equation x2 + y2 = 1 in Zp for p = 2, 3, 5, 13. How many are there in each case?