Take-Home Final Exam
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.
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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)) ∼
(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
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
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) ∼
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) ∼
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).
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