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## Homework Statement

(a) Find a primitive root β of F3[x]/(x^2 + 1).

(b) Find the minimal polynomial p(x) of β in F3[x].

(c) Show that F3[x]/(x^2 + 1) is isomorphic to F3[x]/(p(x)).

## The Attempt at a Solution

I am completely lost on this one

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- Thread starter cooljosh2k2
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- #1

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(a) Find a primitive root β of F3[x]/(x^2 + 1).

(b) Find the minimal polynomial p(x) of β in F3[x].

(c) Show that F3[x]/(x^2 + 1) is isomorphic to F3[x]/(p(x)).

I am completely lost on this one

- #2

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Can you tell us what a primitive root is?

Can you find any root (not just primitive) in our field? To find this, take an arbitrary element a+bx in our field. Can you find a, b and n such that [tex](a+bx)^n=1[/tex]?

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- #4

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I have this: Let F be a field, then a nth root is an element x in F such that [tex]x^n=1[/tex] for some [tex]n>0[/tex]. The nth root x is called primitive if there is no k<n such that [tex]x^k=1[/tex].

Does this make any sense to you? Or do you understand something completely different in the term root?

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- #6

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Can you give me the exact definition in your course? I really want to know if we're talking about thesame thing here...

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