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Please solve these exercises, and show your work. Thank you

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

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

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

(a) (6 points) Find all of the roots of the polynomial h(2) = 73 + 4.×2 ++1 € Z7[2]

(b) (10 points) For f(x) = 26 + 325 + 4×2 – 3x + 2 and g(x) = 3×2 + 2x – 3 in Z-[z], find q(2) and

r(x) as described by the division algorithm so that f(x) = g(x)q(2) +r(2). Be sure to reduce your

final answers mod 7.

3. Determine whether the following polynomials are irreducible in Z[2] (Hint: you should be able to prove

these using the methods from lecture in section 23).

(a) (10 points) f(x) = 23 – 82x + 432

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(b) (10 points) g(x) = = 25+ 24 +23 +22 ++1

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4. (15 points) Count the number of irreducible polynomials of degree 3 in the polynomial ring Z5 [2] (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) 0:RxR+C with o(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 & R* and ab € RX

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

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values of a variable

Roots of The Polynomial

the highest exponent

irreducible polynomial

product of two non constant polynomials

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