Show that if $a_{0}+a_{1}x+...+a_{n}x^{n}$ is irreducible in K[X] then $a_{n}+a_{n-1}x+...+a_{0}x^{n}$ is also irreducible.

Question

Let K be a field and let $$ f(x) = a_{0}+a_{1}x+...+a_{n}x^{n} $$ is irreducible in K[X] then $$ a_{n}+a_{n-1}x+...+a_{0}x^{n} $$ is also irreducible.

The answer is an immediate consequence of this answer, so it's essentially a dupe. — Bill Dubuque, Nov 20 '16 at 00:29
Is there a bijection drawn between the factors of the first and the factors of the second? — Jacob Wakem, Nov 20 '16 at 01:22

score 1 · Answer 1 · answered Nov 20 '16 at 00:13

1

If $P$ denotes the first polynomial, then the second is $Q(X) = X^n P(X^{-1})$. Try to use this.

answered Nov 20 '16 at 00:13

Pedro

122,002

Show that if $a_{0}+a_{1}x+...+a_{n}x^{n}$ is irreducible in K[X] then $a_{n}+a_{n-1}x+...+a_{0}x^{n}$ is also irreducible.

1 Answers1