Proving X

pokoljitef2

pokoljitef2

Answered question

2022-06-21

Proving X 4 14 X 2 + 9 is irreducible

Answer & Explanation

lorienoldf7

lorienoldf7

Beginner2022-06-22Added 19 answers

Let d = [ Q ( 2 + 5 ) : Q ]
The fact that 2 + 5 is a root of X 4 14 X 2 + 9 implies that d 4
But one also has
d = [ Q ( 2 + 5 ) : Q ( 2 ) ] × [ Q ( 2 ) : Q ] = 2 × [ Q ( 2 + 5 ) : Q ( 2 ) ]
, and because [ Q ( 2 + 5 ) : Q ( 2 ) ] 2, you get that d 4
So d = 4, so X 4 14 X 2 + 9 is the minimal polynomial of 2 + 5

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