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...

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

Extracted text: 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].

Jun 04, 2022

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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...

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