# basis of ideal in algebraic number field

Theorem. Let ${\mathcal{O}}_{K}$ be the maximal order^{} of the algebraic number field^{} $K$ of degree $n$. Every ideal $\U0001d51e$ of ${\mathcal{O}}_{K}$ has a basis, i.e. there are in $\U0001d51e$ the linearly independent^{} numbers ${\alpha}_{1},{\alpha}_{2},\mathrm{\dots},{\alpha}_{n}$ such that the numbers

$${m}_{1}{\alpha}_{1}+{m}_{2}{\alpha}_{2}+\mathrm{\dots}+{m}_{n}{\alpha}_{n},$$ |

where the ${m}_{i}$’s run all rational integers, form precisely all numbers of $\U0001d51e$. One has also

$$\U0001d51e=({\alpha}_{1},{\alpha}_{2},\mathrm{\dots},{\alpha}_{n}),$$ |

i.e. the basis of the ideal can be taken for the system of generators^{} of the ideal.

Since $\{{\alpha}_{1},{\alpha}_{2},\mathrm{\dots},{\alpha}_{n}\}$ is a basis of the field extension $K/\mathbb{Q}$, any element of $\U0001d51e$ is uniquely expressible in the form ${m}_{1}{\alpha}_{1}+{m}_{2}{\alpha}_{2}+\mathrm{\dots}+{m}_{n}{\alpha}_{n}$.

It may be proven that all bases of an ideal $\U0001d51e$ have the same discriminant^{} $\mathrm{\Delta}({\alpha}_{1},{\alpha}_{2},\mathrm{\dots},{\alpha}_{n})$, which is an integer; it is called the discriminant of the ideal. The discriminant of the ideal has the minimality property, that if ${\beta}_{1},{\beta}_{2},\mathrm{\dots},{\beta}_{n}$ are some elements of $\U0001d51e$, then

$$\mathrm{\Delta}({\beta}_{1},{\beta}_{2},\mathrm{\dots},{\beta}_{n})\geqq \mathrm{\Delta}({\alpha}_{1},{\alpha}_{2},\mathrm{\dots},{\alpha}_{n})\mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}\mathrm{\Delta}({\beta}_{1},{\beta}_{2},\mathrm{\dots},{\beta}_{n})=0$$ |

But if $\mathrm{\Delta}({\beta}_{1},{\beta}_{2},\mathrm{\dots},{\beta}_{n})=\mathrm{\Delta}({\alpha}_{1},{\alpha}_{2},\mathrm{\dots},{\alpha}_{n})$, then also the ${\beta}_{i}$’s form a basis of the ideal $\U0001d51e$.

Example. The integers of the quadratic field $\mathbb{Q}(\sqrt{2})$ are $l+m\sqrt{2}$ with $l,m\in \mathbb{Z}$. Determine a basis $\{{\alpha}_{1},{\alpha}_{2}\}$ and the discriminant of the ideal a) $(6-6\sqrt{2},\mathrm{\hspace{0.17em}9}+6\sqrt{2})$, b) $(1-2\sqrt{2})$.

a) The ideal may be seen to be the principal ideal^{} $(3)$, since the both generators are of the form $(l+m\sqrt{2})\cdot 3$ and on the other side, $3=0\cdot (6-6\sqrt{2})+(3-2\sqrt{2})(9+6\sqrt{2})$. Accordingly, any element of the ideal are of the form

$$({m}_{1}+{m}_{2}\sqrt{2})\cdot 3={m}_{1}\cdot 3+{m}_{2}\cdot 3\sqrt{2}$$ |

where ${m}_{1}$ and ${m}_{2}$ are rational integers. Thus we can infer that

$${\alpha}_{1}=3,{\alpha}_{2}=3\sqrt{2}$$ |

is a basis of the ideal concerned. So its discriminant is

$$\mathrm{\Delta}({\alpha}_{1},{\alpha}_{2})={\left|\begin{array}{cc}\hfill 3\hfill & \hfill 3\sqrt{2}\hfill \\ \hfill 3\hfill & \hfill -3\sqrt{2}\hfill \end{array}\right|}^{2}=648.$$ |

b) All elements of the ideal $(1-2\sqrt{2})$ have the form

$\alpha =(a+b\sqrt{2})(1-2\sqrt{2})=(a-4b)+(b-2a)\sqrt{2}\mathit{\hspace{1em}}\text{with}a,b\in \mathbb{Z}.$ | (1) |

Especially the rational integers of the ideal satisfy $b-2a=0$, when $b=2a$ and thus $\alpha =a-4\cdot 2a=-7a$. This means that in the presentation^{} $\alpha ={m}_{1}{\alpha}_{1}+{m}_{2}{\alpha}_{2}$ we can assume ${\alpha}_{1}$ to be $7$. Now the rational portion $a-4b$ in the form (1) of $\alpha $ should be splitted into two parts so that the first would be always divisible by 7 and the second by $b-2a$, i.e. $a-4b=7{m}_{1}+(b-2a)x$; this equation may be written also as

$$(2x+1)a-(x+4)b=7{m}_{1}.$$ |

By experimenting, one finds the simplest value $x=3$, another would be $x=10$. The first of these yields

$$\alpha =7(a-b)+(b-2a)(3+\sqrt{2})={m}_{1}\cdot 7+{m}_{2}(3+\sqrt{2}),$$ |

i.e. we have the basis

$${\alpha}_{1}=7,{\alpha}_{2}=3+\sqrt{2}.$$ |

The second alternative $x=10$ similarly would give

$${\alpha}_{1}=7,{\alpha}_{2}=10+\sqrt{2}.$$ |

For both alternatives, $\mathrm{\Delta}({\alpha}_{1},{\alpha}_{2})=392$.

Title | basis of ideal in algebraic number field |

Canonical name | BasisOfIdealInAlgebraicNumberField |

Date of creation | 2013-03-22 17:51:15 |

Last modified on | 2013-03-22 17:51:15 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 12F05 |

Classification | msc 11R04 |

Classification | msc 06B10 |

Synonym | basis of ideal in number field |

Related topic | IntegralBasis |

Related topic | IdealNorm |

Related topic | AlgebraicNumberTheory |

Defines | basis of ideal |

Defines | ideal basis |

Defines | discriminant of the ideal |