Ring theory MOC

Algebraic element

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

An element 𝑎 ∈𝐴 is called algebraic over 𝕂 iff there exists a nonzero polynomial 𝑝(𝑥) ∈𝕂[𝑥] such that 𝑝(𝑎) =0. falg An element which is not algebraic is called transcendental over 𝕂. If 𝑎 is algebraic, the solving ^monic of smallest degree 𝑚𝑎(𝑥) ∈𝕂[𝑥] is called the minimal polynomial of 𝑎. This is a special case of Integral element, and thus the set is denoted O𝐴:𝕂

𝐴 is called algebraic over 𝕂 iff every 𝑎 ∈𝐴 is algebraic, and if 𝐴 is a field the field extension 𝐴 :𝕂 is called algebraic.

An algebraic element over ℚ 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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