Algebraic Number

Abstract

An algebraic element is a generalization of the concept of an algebraic number to elements of any field extension. In its most common application, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational coefficients.

Definition

Let $L$ be an extension field of a field $K$. An element $a\in L$ is said to be algebraic over $K$ if there exists a non-zero polynomial $f(x)$ with coefficients in $K$ such that $f(a)=0$.

More formally, $a$ is algebraic over $K$ if the kernel of the evaluation map ${\varphi }_a:K[x]\to L$ is non-trivial, where ${\varphi }_a(f)=f(a)$. If an element is not algebraic, it is called transcendental.

Properties

• Minimal Polynomial: For every algebraic element $a$ over $K$, there exists a unique monic irreducible polynomial $p(x)\in K[x]$ of smallest degree such that $p(a)=0$. This is called the Minimal Polynomial.
• Degree: The degree of an algebraic element is the degree of its minimal polynomial.
• Field Extensions: If $a$ is algebraic over $K$, then the smallest subfield of $L$ containing both $K$ and $a$, denoted $K(a)$, is a finite Vector Space over $K$.
• Algebraic Closure: The set of all elements in $L$ that are algebraic over $K$ forms a subfield of $L$, known as the Algebraic Closure of $K$ in $L$.

Examples

• Rational Roots: Every rational number $q\in \mathbb{Q}$ is algebraic over $\mathbb{Q}$ because it is a root of $x-q=0$.
• Square Roots: The number $\sqrt{2}$ is algebraic over $\mathbb{Q}$ since it satisfies $x^2-2=0$.
• Imaginary Unit: The number $i$ is algebraic over $\mathbb{Q}$ because it satisfies $x^2+1=0$.
• Transcendental Numbers: The numbers $\pi$ and $e$ are known to be transcendental over $\mathbb{Q}$, meaning they are not algebraic.

Related Concepts

• Field Extension: The context in which algebraic elements are defined
• Minimal Polynomial: The unique irreducible polynomial associated with an algebraic element
• Algebraic Number Field: A finite extension of the field of rational numbers
• Transcendental Number: Elements that are not algebraic

References

https://en.m.wikipedia.org/wiki/Algebraic_element