Ring theory MOC

Algebraic element

Let be a field and be a K-monoid (or extension field, see extension field as a unital associative algebra).

An element is called algebraic over iff there exists a nonzero polynomial such that . falg An element which is not algebraic is called transcendental over . If is algebraic, the solving ^monic of smallest degree is called the minimal polynomial of . This is a special case of Integral element, and thus the set is denoted

is called algebraic over iff every is algebraic, and if is a field the field extension is called algebraic.

An algebraic element over is called an algebraic number.

Examples

• Number field
• All elements of a finite-dimensional unital associative algebra are algebraic

Properties

• Subalgebra generated by an algebraic element
• An algebraic element is invertible iff its minimal polynomial has a nonzero constant term
• Roots of a minimal polynomial
• Spectrum of an algebraic element
• Embedding an algebraic extension into an algebraically closed field

Constructions

• Algebraic interior of a field extension

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