To prove f = x 4 + x 3 + x 2 + x 1...

Zion Wheeler

Zion Wheeler

Answered

2022-06-26

To prove f = x 4 + x 3 + x 2 + x 1 + 1 is irreducible over the Q

Answer & Explanation

benedictazk

benedictazk

Expert

2022-06-27Added 22 answers

The polynomial is reciprocal and can be easily factored over the reals:
x 4 + x 3 + x 2 + x + 1 = x 2 ( ( x 2 + 1 x 2 ) + ( x + 1 x ) + 1 ) = x 2 ( ( x + 1 x ) 2 + ( x + 1 x ) 1 ) = x 2 ( x + 1 x 1 + 5 2 ) ( x + 1 x 1 5 2 ) = ( x 2 + 1 5 2 x + 1 ) ( x 2 + 1 + 5 2 x + 1 )
Since neither quadratic has real roots this is the unique irreducible factorization over R , and since the coefficients are not rational the polynomial is irreducible over Q

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