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'irreducible polynomial'
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| Title of object: |
irreducible polynomial |
| Canonical Name: |
IrreduciblePolynomial2 |
| Type: |
Definition |
| Created on: |
2004-06-10 04:12:31 |
| Modified on: |
2005-03-06 17:11:09 |
| Classification: |
msc:12D10 |
| Keywords: |
irreducible, indivisible |
| Synonyms: |
irreducible polynomial=prime polynomial
irreducible polynomial=indivisible polynomial |
Preamble:
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Content:
Let \,$f(x) = a_0+a_1x+...+a_nx^n$\, 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 $\mathbb{Q}(a_0, a_1, ..., a_n)$, then the polynomial $f(x)$ is said to be \PMlinkescapetext{{\em irreducible}}.
\textbf{Examples.} \,All linear polynomials are \PMlinkescapetext{irreducible}. \, The polynomials $x^2-3$, $x^2+1$ and $x^2-i$ are \PMlinkescapetext{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 \PMlinkescapetext{irreducible}. |
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