irreducible polynomial

Math 120B Name: Spring 2021 Take-Home Final Exam Student ID: Due Mon 6/9/2021 at 11:59pm in Canvas • There are 7 questions for a total of 100 points. • Some questions have several parts. • For full credit show ALL of your work, explain your process fully. Make sure that I can understand exactly HOW you got your answer. • Please box your final answer, when applicable. Question: 1 2 3 4 5 6 7 Total Points: 10 10 20 14 16 14 16 100 Score: Page 1 of 2 √ √   √ √   1. (10 points) Find a basis of the field extension Q 3, 7 : Q, and find the degree Q 3, 7 : Q .   2. (10 points) Find a value γ ∈ R such that Q 51/2 , 51/3 = Q(γ), and prove your assertion. 3. Find an example in each of the following cases: (a) (10 points) Find an example of a subring of Z[x] which is not an ideal of Z[x]. (b) (10 points) Find an example of a finite, non-commutative ring. 4. (a) (8 points) Find a monic, irreducible polynomial m(x) ∈ Z2 [x] such that Z2 [x]/(m(x)) ∼ = F8 (b) (6 points) By part (a) and Kronecker’s theorem, we know Z2 [x]/(m(x)) ∼ = Z2 [β] ∼ = F8 , where ( ) x + m(x) 7−→ β Show Z2 [β 2 ] = Z2 [β]. Hint: you can use Tower Law. 5. Consider the element α = 21/3 + 21/2 ∈ R. (a) (8 points) Find the minimal polynomial for α over Q(21/2 ). Specifically, find m(x) ∈ Q(21/2 )[x] Hint: you may use Tower Law to prove irreducibility. (b) (8 points) Find the minimal polynomial for α over Q. Specifically, find m(x) ∈ Q[x]. You may use your work from part (a) to help justify irreducibility. 6. Determine whether the following statements are true or false. Justify with a proof or a counterexample. (a) (7 points) If P and Q are both prime ideals of R, then P ∩ Q is a prime ideal of R. (b) (7 points) If F is a field and F is the algebraic closure of F , there is an example where [F : F ] is finite. 7. The First Isomorphism Theorem has two important corollaries: the Second Isomorphism Theorem and the Third Isomorphism Theorem. For this exam, we will investigate the Third Isomorphism Theorem for rings: Theorem 0.1. (Third Isomorphism Theorem for rings) Let I and J be ideals of ring R, with I ⊆ J. Then I is an ideal of J, and (R/I) (J/I) ∼ = R/J / Note that this theorem allows us to greatly simplify cases where we would construct a factor ring out of another factor ring. For example, (Z/18Z) (9Z/18Z) ∼ = Z/9Z / This question will walk you through the steps for the proof of the Third Isomorphism Theorem. Consider the homomorphism φ : R/I −→ R/J with φ(a+I) = a+J. (I will allow you to assume φ is a well-defined homomorphism on this exam). (a) (4 points) Prove I is an ideal of J. (This allows us to define J/I) (b) (4 points) Prove that ker φ = J/I. (c) (4 points) Prove φ(R/I) = R/J, i.e. prove φ is onto. (d) (4 points) Use the First Isomorphism Theorem on φ to prove the Third Isomorphism Theorem. (Using parts (a),(b), and (c), you are able to write part (d) with only one or two lines of proof). Page 2 of 2
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