Olympiad Proof Problem - Day 5

Prove that for all for primes $p$ the polynomial

$x^{p-1} + x^{p-2} + x^{p-3} + \ldots + 1$

is irreducible over rational numbers.

Try more proof problems at Olympiad Proof Problems.

#Olympiad #Proof #ProofProblems #SuryaPrakash #OlympiadProofProblems

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

I used Eisenstein's Irreducibility Criterion to solve this question.

Let

$\Phi_p (x) = x^{p-1} + x^{p-2} + \ldots + x^2 + x + 1 = \dfrac {x^p - 1}{x-1}$

We claim that $\Phi_p (x)$ is irreducible over the rational numbers. Although $\Phi_p (x)$ itself does not directly admit application of Eisenstein's criterion, a minor variant of it does. That is, consider

$f(x) = \Phi_p (x+1) = \dfrac {(x+1)^p - 1}{(x+1)-1} = \dfrac {\displaystyle \sum_{k=0}^{p-1} {p \choose k} x^{p-k}}{x}$

$=x^{p-1}+{p \choose 1} x^{p-2} + {p \choose 2} x^{p-3} + \ldots + {p \choose {p-2}} x + {p \choose {p-1}}$

All the lower coefficients are divisible by $p$ , and the constant coefficient is exactly $p$ , so it is not divisible by $p^2$ . Thus, Eisenstein's criterion applies, and $f$ is irreducible. Certainly, if $\Phi_p (x) = g(x) h(x)$ then $f(x) = \Phi_p (x+1) = g(x+1) h(x+1)$ gives a factorisation of $f$ . Thus, $\Phi_p$ has no proper factorisation, i.e. it is irreducible.

Sharky Kesa - 5 years, 7 months ago

Nice solution. But why don't you do it another way or rather prove Eisenstein's criterion?

Surya Prakash - 5 years, 7 months ago

I have written in the proof in the wiki.

Sharky Kesa - 5 years, 7 months ago

Well, is cyclotomic polynomials allowed?

Pi Han Goh - 5 years, 7 months ago

@Pi Han Goh – Yeah, sure. Post your solution using cyclotomic polynomials.

Sharky Kesa - 5 years, 7 months ago

When is Day 6 going to be posted?

Sharky Kesa - 5 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$