Proving that ( x 2 </msup> + 2 ) n </msup> + 5 ( x

Carly Cannon

Carly Cannon

Answered question

2022-07-07

Proving that ( x 2 + 2 ) n + 5 ( x 2 n 1 + 10 x 2 + 5 ) is irreducible

Answer & Explanation

lofoptiformfp

lofoptiformfp

Beginner2022-07-08Added 16 answers

Using the criterion you mention, the only thing left to show is that x 2 + 2 does not divide x 2 n 1 + 10 x 2 + 5 over the field F 5 . (This is just a fancy way of saying "modulo 5".)
However, note that modulo 5, we have
x 2 n 1 + 10 x 2 + 5 x 2 n 1 .
Since F 5 [ x ] is a UFD, we can conclude directly that x 2 + 2 does not divide x 2 n 1 and this finishes the job.

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