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[parent] irreducible polynomial (Definition)

Let $ f(x) = a_0\!+\!a_1x\!+\cdots+\!a_nx^n$ be a polynomial with complex coefficients $ a_{\nu}$ and with the degree $ n > 0$. If $ f(x)$ can not be written as product of two polynomials with positive degrees and with coefficients in the field $ \mathbb{Q}(a_0,\,a_1,\,\ldots,\,a_n)$, then the polynomial $ f(x)$ is said to be irreducible. Otherwise, $ f(x)$ is reducible.

Examples. All linear polynomials are irreducible. The polynomials $ x^2\!-\!3$, $ x^2\!+\!1$ and $ x^2\!-\!i$ are irreducible (although they split in linear factors in the fields $ \mathbb{Q}(\sqrt{3})$, $ \mathbb{Q}(i)$ and $ \mathbb{Q}(\frac{1\!+\!i}{\sqrt{2}})$, respectively). The polynomials $ x^4\!+\!4$ and $ x^6\!+\!1$ are not irreducible.

The above definition of irreducible polynomial is special case of the more general setting where $ f(x)$ is a non-constant polynomial in the polynomial ring $ K[x]$ of a field $ K$; if $ f(x)$ is not expressible as product of two polynomials with positive degrees in the ring $ K[x]$, then $ f(x)$ is irreducible (in $ K[x]$).

Example. If $ K$ is the Galois field with two elements (0 and 1), then the trinomial $ x^2\!+\!x\!+\!1$ of $ K[x]$ is irreducible (because an equation $ x^2\!+\!x\!+\!1 = (x\!+\!a)(x\!+\!b)$ would imply the two conflicting conditions $ a\!+\!b = 1$ and $ ab = 1$).



"irreducible polynomial" is owned by pahio.
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See Also: Eisenstein criterion, irreducible, monic

Other names:  prime polynomial, indivisible polynomial
Also defines:  irreducible polynomial, reducible
Keywords:  irreducible

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irreducibility of binomials with unity coefficients (Result) by pahio
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Cross-references: imply, equation, trinomial, Galois field, ring, expressible, polynomial ring, factors, field, degrees, positive, product, coefficients, complex, polynomial
There are 41 references to this entry.

This is version 15 of irreducible polynomial, born on 2004-06-10, modified 2007-12-28.
Object id is 5907, canonical name is IrreduciblePolynomial2.
Accessed 10576 times total.

Classification:
AMS MSC: 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )
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Odd browser fault by pahio on 2006-01-05 14:03:22
Hi experts! My Mozilla Firefox (1.5) has behaved for 2 weeks oddly: it does not show any PM entry (except their TeX sources) at all; the same concerns the MS Exploder on my computer. I cannot fix the fault. Please tell what is the name of the fault and how I could fix it.
Quite funny that my Firefox can percectly show the PlanetPhysics entries!
Regards,
Jussi
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field notation by mathforever on 2005-04-16 07:46:12
I have quite a remote idea what the notation

$\mathbb{Q}(a_0, a_1, ..., a_n)$

actually mean. So, could you possibly write that explicitly? I don't know how frequent is its ussage on PM, but if it is frequent then may be it is a good idea to make a separate entry with explanation, and make a link to it here.

Serg.
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