Absolute Discriminant

Definition

The absolute discriminant is a mathematical invariant associated with a number field that provides essential information about the field's structure and properties. It is derived from the discriminant of the field's ring of integers and plays a crucial role in understanding the arithmetic properties, including the ramification of primes and the behavior of integral elements within the field. The absolute discriminant is particularly important when calculating various properties of algebraic extensions and can be used to distinguish between different number fields.

5 Must Know Facts For Your Next Test

1. The absolute discriminant can be calculated as the absolute value of the discriminant of a basis for the ring of integers of a number field.
2. If the number field is generated by a root of an irreducible polynomial of degree n, then the absolute discriminant is related to the determinant of a specific matrix associated with that polynomial.
3. The absolute discriminant helps determine the splitting behavior of primes in the number field; larger values often indicate more complex ramification.
4. It is closely linked to various arithmetic properties such as class number, regulator, and the structure of units within the number field.
5. For quadratic fields, the absolute discriminant can be expressed simply as $$D = d^2 - 4ac$$ for a polynomial $$ax^2 + bx + c$$.

Review Questions

• How does the absolute discriminant relate to the properties of a number field's ring of integers?
  • The absolute discriminant is calculated from the discriminant of the ring of integers in a number field and reveals important aspects about its structure. It provides insights into how prime ideals split or ramify in that field, affecting both arithmetic and geometric interpretations. A deeper understanding of this relationship allows for better comprehension of how elements behave under various operations within the number field.
• Discuss how changes in the absolute discriminant impact the classification of number fields.
  • Changes in the absolute discriminant can significantly impact how number fields are classified. For instance, when comparing two number fields, if their absolute discriminants differ, they may exhibit distinct ramification behavior or varying properties related to their rings of integers. This classification aids mathematicians in identifying whether certain fields are isomorphic or whether they have unique arithmetic properties that distinguish them from others.
• Evaluate the implications of having a large absolute discriminant on the arithmetic properties of a number field.
  • A large absolute discriminant often indicates that the corresponding number field has complicated ramification and intricate relationships between its prime ideals. This complexity can lead to higher class numbers, which suggest more challenging structures regarding units and factorization. Understanding these implications helps mathematicians investigate further into not only the properties of specific fields but also their connections to broader theories within algebraic number theory.

Related terms

Discriminant: A value computed from a polynomial that gives information about the nature of its roots, specifically whether they are distinct or repeated.

Number Field: A finite degree field extension of the rational numbers, which can be represented as $$\mathbb{Q}(\alpha)$$ for some algebraic element $$\alpha$$.

Ring of Integers: The integral closure of the integers in a number field, consisting of all elements that are roots of monic polynomials with integer coefficients.