# irreducible polynomial

Let $f(x)={a}_{0}+{a}_{1}x+\mathrm{\cdots}+{a}_{n}{x}^{n}$ be a polynomial^{} with complex coefficients ${a}_{\nu}$ and with the degree (http://planetmath.org/Polynomial) $n>0$. If $f(x)$ can not be written as product of two polynomials with positive degrees and with coefficients in the field $\mathbb{Q}({a}_{0},{a}_{1},\mathrm{\dots},{a}_{n})$, then the polynomial $f(x)$ is said to be . Otherwise, $f(x)$ is reducible.

Examples. All linear polynomials are . The polynomials ${x}^{2}-3$, ${x}^{2}+1$ and ${x}^{2}-i$ are (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 .

The above definition of 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 (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 (because an equation ${x}^{2}+x+1=(x+a)(x+b)$ would imply the two conflicting conditions $a+b=1$ and $ab=1$).

Title | irreducible polynomial |

Canonical name | IrreduciblePolynomial |

Date of creation | 2013-03-22 14:24:22 |

Last modified on | 2013-03-22 14:24:22 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 18 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 12D10 |

Synonym | prime polynomial |

Synonym | indivisible polynomial |

Related topic | EisensteinCriterion |

Related topic | Irreducible |

Related topic | Monic2 |

Defines | irreducible polynomial |

Defines | reducible |