# discriminant

## 1 Definitions

Let $R$ be any Dedekind domain^{} with field of fractions^{} $K$. Fix a finite dimensional field extension $L/K$ and let $S$ denote the integral closure^{} of $R$ in $L$. For any basis ${x}_{1},\mathrm{\dots},{x}_{n}$ of $L$ over $K$, the determinant

$$\mathrm{\Delta}({x}_{1},\mathrm{\dots},{x}_{n}):=det[\mathrm{Tr}({x}_{i}{x}_{j})],$$ |

whose entries are the trace of ${x}_{i}{x}_{j}$ over all pairs $i,j$, is called the discriminant^{} of the basis ${x}_{1},\mathrm{\dots},{x}_{n}$. The ideal in $R$ generated by all discriminants of the form

$$\mathrm{\Delta}({x}_{1},\mathrm{\dots},{x}_{n}),{x}_{i}\in S$$ |

is called the discriminant ideal of $S$ over $R$, and denoted $\mathrm{\Delta}(S/R)$.

In the special case where $S$ is a free $R$–module, the discriminant ideal $\mathrm{\Delta}(S/R)$ is always a principal ideal^{}, generated by any discriminant of the form $\mathrm{\Delta}({x}_{1},\mathrm{\dots},{x}_{n})$ where ${x}_{1},\mathrm{\dots},{x}_{n}$ is a basis for $S$ as an $R$–module. In particular, this situation holds whenever $K$ and $L$ are number fields^{}.

## 2 Alternative notations

The discriminant is sometimes denoted with $\mathrm{disc}$ instead of $\mathrm{\Delta}$. In the context of number fields, one often writes $\mathrm{disc}(L/K)$ for $\mathrm{disc}({\mathcal{O}}_{L}/{\mathcal{O}}_{K})$ where ${\mathcal{O}}_{L}$ and ${\mathcal{O}}_{K}$ are the rings of algebraic integers of $L$ and $K$. If $K$ or ${\mathcal{O}}_{K}$ is omitted, it is typically assumed to be $\mathbb{Q}$ or $\mathbb{Z}$.

## 3 Properties

The discriminant is so named because it allows one to determine which ideals of $R$ are ramified in $S$. Specifically, the prime ideals^{} of $R$ that ramify in $S$ are precisely the ones that contain the discriminant ideal $\mathrm{\Delta}(S/R)$. In the case $R=\mathbb{Z}$, a theorem of Minkowski (http://planetmath.org/MinkowskisConstant) that any ring of integers $S$ of a number field larger than $\mathbb{Q}$ has discriminant strictly smaller than $\mathbb{Z}$ itself, and this fact combined with the previous result shows that any number field $K\ne \mathbb{Q}$ admits at least one ramified prime over $\mathbb{Q}$.

## 4 Other types of discriminants

In the special case where $L=K[x]$ is a primitive separable field extension of degree $n$, the discriminant $\mathrm{\Delta}(1,x,\mathrm{\dots},{x}^{n-1})$ is equal to the polynomial discriminant (http://planetmath.org/PolynomialDiscriminant) of the minimal polynomial $f(X)$ of $x$ over $K[X]$.

The discriminant of an elliptic curve^{} can be obtained by taking the polynomial^{} discrimiant of its Weierstrass polynomial, and the modular discriminant^{} of a complex lattice equals the discriminant of the elliptic curve represented by the corresponding lattice quotient.

Title | discriminant |
---|---|

Canonical name | Discriminant1 |

Date of creation | 2013-03-22 12:37:57 |

Last modified on | 2013-03-22 12:37:57 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 12 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 11R29 |

Related topic | IntegralBasis |

Related topic | PolynomialDiscriminant |

Related topic | ModularDiscriminant |

Defines | discriminant ideal |