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Hi, here is my paper, I need help to finish these questions. There are totally 6 questions, you will have 2 days to finish this paper. Thank you.

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Math 120B

Name:

Spring 2021

Take-Home Midterm Exam

Student ID:

Due Mon 5/12/2020 at 11:59pm in Canvas

• There are 6 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

Total

Points:

18

16

20

15

21

10

100

Score:

Page 1 of 2

1. (a) (8 points) Find the remainder of 51011 modulo 303

(b) (10 points) Find all x ∈ Z solutions to 155x ≡ 75 mod 65, if any exist.

2. Consider the following problems in Z7 [x]:

(a) (6 points) Find all of the roots of the polynomial h(x) = x3 + 4×2 + x + 1 ∈ Z7 [x]

(b) (10 points) For f (x) = x6 + 3×5 + 4×2 − 3x + 2 and g(x) = 3×2 + 2x − 3 in Z7 [x], find q(x) and

r(x) as described by the division algorithm so that f (x) = g(x)q(x) + r(x). Be sure to reduce your

final answers mod 7.

3. Determine whether the following polynomials are irreducible in Z[x] (Hint: you should be able to prove

these using the methods from lecture in section 23).

(a) (10 points) f (x) = x3 − 82x + 432

(b) (10 points) g(x) =

x6 − 1

= x5 + x4 + x3 + x2 + x + 1

x−1

4. (15 points) Count the number of irreducible polynomials of degree 3 in the polynomial ring Z5 [x] (Hint:

you do not have to list every polynomial to make your argument formal).

5. Determine whether the following statements are true or false. Justify with a proof or a counterexample.

(a) (7 points) φ : R × R −→ C with φ((a, b)) = a + bi is an isomorphism.

(b) (7 points) 3Z/9Z ∼

= Z3 as rings.

(c) (7 points) For a ring R, it is possible to have a, b 6∈ R× and ab ∈ R×

6. (10 points) Give an example of a non-commutative ring of characteristic 2, or prove that none exists.

Page 2 of 2

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Math Problems

negative numbers

positive numbers

Non commutative ring

Polynomials numbers

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