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Are there any other tests other than Eisenstein's general irreducibility test . And i wonder if it is possible to extend it to more than one variable . looking forward to get some response.

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3 Answers 3

Much more powerful techniques for deducing irreducibility and factorization properties arise by applying valuation-theoretic concepts embodied in the notion of a Newton Polygon. For some references see this answer.

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There are well known : Cohn's irreducibility criterion and irreducibility criterion of Perron .

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For a test closely related to Eisenstein's:

There was a nice article by David A. Cox in the Monthly ("Why Eisenstein proved the Eisenstein Criterion and why Schönemann discovered it first", Volume 118 no 1 (January 2011), pages 3-21). It gives:

Schönemann's Irreducibility Criterion. Let $f(x)$ be a polynomial with integer coefficients and positive degree. Assume there is a prime $p$ and an integer $a$ such that $$f(x) = (x-a)^n + pF(x),\qquad F(x)\in\mathbb{Z}[x];$$ if $F(a)\not\equiv 0 \pmod{p}$, then $f(x)$ is irreducible modulo $p^2$.

The test implies Eisenstein's criterion: say $f(x) = a_nx^n+\cdots +a_0$ with integer coefficients, $\gcd(a_n,p)=1$, $a_i\equiv 0\pmod{p}$ for $i=0,1,\ldots,n-1$, and $a_0\not\equiv 0\pmod{p^2}$. Multiply by a suitable integer so that $a_n\equiv 1\pmod{p}$; then we can write $$f(x) = x^n + pF(x)$$ with $F(x)\in \mathbb{Z}[x]$, and $F(0)\not\equiv 0\pmod{p}$. Hence $f(x)$ is irreducible modulo $p^2$, and hence over $\mathbb{Z}$.

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