Norm

Definition

In algebraic number theory, the norm of an algebraic number is a value that gives important information about its behavior in relation to a field extension. It can be viewed as a multiplicative measure that reflects how the number scales when considered within its minimal field, connecting properties of elements with their corresponding fields and extensions.

5 Must Know Facts For Your Next Test

1. The norm of an algebraic number $eta$ over a field $K$ is calculated as the product of all its conjugates, which are obtained by applying all embeddings of the field into the complex numbers.
2. In number fields, the norm function is multiplicative, meaning that for two elements $eta_1$ and $eta_2$, the norm satisfies $N(eta_1 eta_2) = N(eta_1) N(eta_2)$.
3. The norm plays a crucial role in determining whether an ideal is principal, as norms can be used to define class numbers and analyze the structure of the ideal class group.
4. For algebraic integers, the norm is always an integer, and if an element has a norm equal to 1, it often indicates that it could be a unit in the ring of integers.
5. Norms are used to derive Minkowski bounds, which are essential in establishing finiteness results for ideal class groups in algebraic number theory.

Review Questions

• How does the calculation of the norm for an algebraic number reflect its properties in a given field?
  • The norm for an algebraic number provides insight into how it behaves under various embeddings into complex numbers. Specifically, by calculating the norm as the product of all conjugates, you can see how this number interacts with its minimal polynomial and deduce properties such as whether it belongs to certain types of fields or extensions. This relationship also helps identify whether an ideal generated by such numbers could be principal.
• Discuss how norms contribute to understanding class numbers and properties of ideals in number fields.
  • Norms are vital for exploring class numbers because they allow us to classify ideals based on their size and multiplicative properties. By analyzing norms, we can determine if certain fractional ideals are equivalent or if they generate principal ideals. This investigation directly links norms with class group structure and enables us to apply Minkowski bounds, which further help in assessing whether class numbers are finite.
• Evaluate the role of norms in establishing relationships between discriminants and field extensions.
  • Norms serve as a bridge between discriminants and field extensions by linking their respective mathematical structures. The relationship between these two concepts helps analyze how changing fields impacts various algebraic properties. For instance, knowing how to compute norms allows one to derive discriminants from polynomials, revealing crucial information about Galois groups and ramification behaviors in extensions. Consequently, this understanding deepens insights into both number theory and algebraic geometry.

Related terms

Trace: The trace of an algebraic number is the sum of its conjugates, providing insights into the behavior of the number in the context of its field.

Discriminant: The discriminant is a value derived from a polynomial that gives information about the nature of its roots and is closely related to the properties of the number field.

Integral Basis: An integral basis is a set of algebraic integers in a number field that generates the ring of integers, helping to define the structure of ideals and norms within that field.