How to prove that the polynomial x n </msup> + x <mrow class="MJX-TeXAtom

Abram Boyd

Abram Boyd

Answered question

2022-06-12

How to prove that the polynomial x n + x n 1 + + x 2 + x 1 is irreducible over Q ?

Answer & Explanation

sleuteleni7

sleuteleni7

Beginner2022-06-13Added 28 answers

You can also use the following criterion by Brauer on the (negative) reciprocal
x n f ( 1 / x ) = x n x n 1 1.
Let a 1 a 2 a n be positive integers and n 2. Then the polynomial p ( x ) = x n a 1 x n 1 a 2 x n 2 a n is irreducible over Z
For the proof see for example Prasolov's Polynomials

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