Minimal Polynomial

Definition

The minimal polynomial of an algebraic element over a field is the monic polynomial of smallest degree that has the element as a root. This polynomial captures the essence of the element's algebraic properties and relates closely to the structure of number fields, field extensions, and their algebraic closures.

5 Must Know Facts For Your Next Test

1. The minimal polynomial is unique for each algebraic element over a given field and is irreducible over that field.
2. The degree of the minimal polynomial corresponds to the dimension of the field extension generated by the algebraic element.
3. If an algebraic element is adjoined to a field, its minimal polynomial can be used to construct a new field containing the element.
4. The roots of the minimal polynomial generate a splitting field, which is crucial in understanding normal extensions.
5. Computing the minimal polynomial involves finding the monic polynomial with integer coefficients that has the element as a root and is irreducible over its base field.

Review Questions

• How does the concept of minimal polynomials relate to algebraic elements and their corresponding field extensions?
  • Minimal polynomials are directly tied to algebraic elements since they provide the simplest polynomial equation that these elements satisfy. The degree of the minimal polynomial indicates how many dimensions are added to the original field when we create a field extension by adjoining the algebraic element. In essence, understanding minimal polynomials helps us grasp how field extensions can be formed and what structure they bring to number fields.
• Discuss how minimal polynomials contribute to identifying normal extensions and splitting fields within number theory.
  • Minimal polynomials play a crucial role in identifying normal extensions since they allow us to see if an extension contains all roots of its generating polynomials. When we take an algebraic element and find its minimal polynomial, we can then determine if adjoining this element will yield a splitting field—where all roots are contained—leading to normality. This connection emphasizes how minimal polynomials help us understand both the roots of polynomials and their extensions.
• Evaluate the significance of minimal polynomials in determining properties such as norm and trace maps within field extensions.
  • Minimal polynomials significantly influence calculations related to norm and trace maps, as these maps are defined based on the roots of these polynomials. The norm of an element is related to the product of its roots (which are derived from the minimal polynomial), while the trace corresponds to their sum. By analyzing these relationships, we gain insights into how elements behave in different fields, helping us further explore their structural properties and interrelations within algebraic number theory.

Related terms

Algebraic Element: An element that is a root of a non-zero polynomial with coefficients in a given field.

Degree of a Polynomial: The highest power of the variable in a polynomial expression, which indicates the number of roots it can have in its algebraic closure.

Irreducible Polynomial:

A polynomial that cannot be factored into polynomials of lower degree with coefficients in the same field.