Algebraic dependence

Definition

Algebraic dependence refers to a situation in which a set of elements, typically numbers or variables, satisfies a polynomial equation with coefficients from a certain field. In the context of algebraically closed fields, this concept is significant as it connects to the idea that every non-constant polynomial has roots in the field, ensuring that algebraically dependent elements can be expressed through each other using polynomial relations.

5 Must Know Facts For Your Next Test

1. In an algebraically closed field, any finite set of elements can be shown to be algebraically dependent if there exists a non-trivial polynomial relation among them.
2. Algebraic dependence helps in understanding the structure of algebraically closed fields, as all elements can be expressed in terms of a finite number of algebraically independent ones.
3. If a set of elements is algebraically independent over a field, then any polynomial relation between them must have all coefficients equal to zero.
4. The notion of algebraic dependence is crucial for defining the dimension of an algebraic variety, as it determines the relationships between points in the variety.
5. In the context of field extensions, algebraic dependence plays a key role in determining whether an element can be considered algebraic or transcendental over the base field.

Review Questions

• How does algebraic dependence relate to the concept of algebraically closed fields?
  • Algebraic dependence is closely tied to algebraically closed fields because, in such fields, every non-constant polynomial must have roots. This means that if a set of elements is algebraically dependent, they can be described using polynomial relationships with coefficients from the field. In essence, algebraically closed fields allow us to fully explore these relationships and confirm that every possible dependency among elements can be realized through some polynomial equation.
• Discuss the implications of a set being algebraically independent in relation to the concept of minimal polynomials.
  • When a set of elements is algebraically independent, it means there are no non-trivial polynomial relations among them. This independence directly influences their minimal polynomials since each element's minimal polynomial must be unique and irreducible over the base field. If any element in the set were algebraically dependent on others, its minimal polynomial would not be minimal anymore, as it could be expressed using the others' polynomials, indicating a redundancy in its definition.
• Evaluate how the concepts of algebraic dependence and transcendental elements affect the understanding of field extensions.
  • Algebraic dependence and transcendental elements play a pivotal role in field extensions by determining whether new elements introduced to a field can be classified as algebraic or transcendental. When an element is shown to be algebraically dependent on existing elements, it fits into the structure defined by those elements and can thus be incorporated into the extension naturally. In contrast, transcendental elements are those that do not satisfy any polynomial equation over the base field, indicating a broader complexity in the relationship between different fields and their extensions. This distinction influences how we analyze and understand various properties and behaviors within these extended fields.

Related terms

Algebraically closed field: A field in which every non-constant polynomial equation has at least one root in that field.

Transcendental elements: Elements that are not algebraic over a field, meaning they do not satisfy any polynomial equation with coefficients from that field.

Minimal polynomial: The monic polynomial of least degree that has a given element as a root and whose coefficients belong to a specified field.