Let be a field and . Prove that is a field if and only if is irreductible.

How can I build a field with nine elements?

Prove that

Thanks!

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- Sep 27th 2008, 04:14 PMroporteQuotient ring, field, irreductible
Let be a field and . Prove that is a field if and only if is irreductible.

How can I build a field with nine elements?

Prove that

Thanks! - Sep 27th 2008, 04:31 PMThePerfectHacker
Say that is irreducible then we need to show for any with (i.e. non-zero) we can find such that . Now since are relatively prime since is irreducible it means there exists so that and so i.e. .

Now you try proving the converse by assuming that is reducible.

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How can I build a field with nine elements?

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Prove that