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$5x^{n}-1 \in \mathbb{Q}[x]$ for $n \geq 1$ is irreducible.

I tried to prove it with Eisenstein's Criterion, but I did not know how to use it.

I also used the fact that a polynomial f(x) over a field k is irreducible if the polynomial f(x+1) is irreducible, and so I tried to make the criterion with the expansion of $(x+1)^{n}$, but I got stuck.

could you help me, or give me some clue how to do it?

Sil
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The πzza man
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1 Answers1

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Proposition: If $ f(x)\neq x $ then $f(x)$ is irreducible if and only if its reflected polynomial $f^*(x)$ is irreducible.

Definition: If $f(x)=a_n x^n+\cdots +a_0$,then $f^*(x)=a_n + a_{n-1}x +\cdots +a_1 x^{n-1}+a_0 x^n$

So, $f(x)=5x^n-1 \Rightarrow f^*(x)=-x^n+5$.we can prove $f^*(x)$ is irreducible,hence $f(x)$ is also.

Well
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  • Exactly the argument I would use. Plus 1. But I do worry that OP may not have enough experience in basic algebra to see why your argument is correct. – Lubin Sep 17 '21 at 02:06