# ring without irreducibles

An integral domain^{} may not any
irreducible elements^{}. One such example is the ring of all
algebraic integers^{}. Any nonzero non-unit $\vartheta $ of this ring satisfies an equation

$${x}^{n}+{a}_{1}{x}^{n-1}+\mathrm{\cdots}+{a}_{n-1}x+{a}_{n}=0$$ |

with integer coefficients ${a}_{j}$, since it is an algebraic integer; moreover, we can assume that ${a}_{n}=\text{N}(\vartheta )\ne \pm 1$ (see norm and trace of algebraic number: 2). The element $\vartheta $ has the

$$\vartheta =\sqrt{\vartheta}\cdot \sqrt{\vartheta}.$$ |

Here, $\sqrt{\vartheta}$ belongs to the ring because it satisfies the equation

$${x}^{2n}+{a}_{1}{x}^{2n-2}+\mathrm{\cdots}+{a}_{n-1}{x}^{2}+{a}_{n}=0,$$ |

and it is no unit. Thus the element $\vartheta $ is not irreducible.

Title | ring without irreducibles |
---|---|

Canonical name | RingWithoutIrreducibles |

Date of creation | 2014-05-29 11:39:19 |

Last modified on | 2014-05-29 11:39:19 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 15 |

Author | pahio (2872) |

Entry type | Example |

Classification | msc 13G05 |

Related topic | FieldOfAlgebraicNumbers |