Abstract Algebra Ring Theory Problems

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Reference no: EM132400077

Assignment - Exercise about Abstract Algebra Ring Theory

1. (a) Can you deduce if 2x + 1 is invertible in Z₃[x]/(x² + 2x + 2)? In case of a positive answer, give its inverse.

(b) Can you deduce if x + 2 is invertible in Z₃[x]/(x² + 2x + 2)? In case of a positive answer, give its inverse.

2. (a) How many monic polynomials are there in Z₃[x] of degree2?

(b) What are the monic irreducible polynomials of Z₃[x] of degree 2? Explain your answer.

(c) Choose one of the irreducible polynomials of the previous section and call it f. How many elements are there in Z₃/(f)?

(d) How many elements are there in Z₉?

(e) How many invertible elements are there in Z₃/(f)? Give an invertible element of Z₃/(f) different than 1 and give its inverse.

(f) How many invertible elements are there in Z₉? Give an invertible element of Z₉ different than 1 and give its inverse.

(g) Give a primitive element of Z₃/(f) and write all its different powers in polynomial form.

(h) Give a primitive element of Z₉ and write all its different powers.

Reference no: EM132400077

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