Let f(x) = x² +1 € Z3[x] and let R= Z3[x]/I, where I = (f(x)). (a) Show that R is a field with 9 elements. (b) Denote by 0 := 0 + I, 1 := 1 +I, and a := x +I. Write the other 6 elements of R in terms of a and determine the multiplicative inverse of each nonzero element. (c) Prove that R=Z3[i].

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Q: be a field and let c,d e F. Show that c · (-d) = -(c d).

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Q: Find the splitting field x4 - x2 - 2 over Z3.

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Q: (b) How many elements are there in S? Justify your answer.

A: The elements areNote: the elements of  are

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Q: Consider the field Z3[x]/. Please explain to me how this field has 9 elements. Thank you!

A: Consider the field Z3[x]/<x2 +x+2>. Please explain to me how this field has 9 elements.

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