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Find the remainder of 51011 modulo 303 (b) (10 points) Find all r e Z solutions to 155x = 75 mod 65, if any exist

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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.x2 ++1 € Z7[2] (b) (10 points) For f(x) = 26 + 325 + 4x2 – 3x + 2 and g(x) = 3x2 + 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 + 432 226 - 1 (b) (10 points) g(x) = = 25+ 24 +23 +22 ++1 1 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.