Algebraic extension

In abstract algebra, an algebraic extension is a field extension $L/K$ such that every element of the larger field $L$ is algebraic over the smaller base field $K$, meaning every element of $L$ is a root of some non-zero polynomial with coefficients in $K$.

Definition and Basic Concepts

Let $L/K$ be a field extension. An element $\alpha \in L$ is said to be algebraic over $K$ if there exists a non-zero polynomial $p(x) \in K[x]$ such that $p(\alpha) = 0$. If an element of $L$ is not algebraic over $K$, it is called transcendental over $K$.

A field extension $L/K$ is classified as an algebraic extension if every single element of $L$ is algebraic over $K$. Conversely, if there exists at least one element in $L$ that is transcendental over $K$, the extension is called a transcendental extension. The study of algebraic extensions forms a central pillar of field theory and Galois theory, providing the framework for understanding the roots of polynomials and the symmetries of field structures.

Finite vs. Infinite Algebraic Extensions

A field extension $L/K$ is called a finite extension if $L$ is a finite-dimensional vector space over $K$. The dimension of $L$ as a $K$-vector space is called the degree of the extension, denoted by $[L:K]$.

A fundamental theorem in field theory states that every finite extension is an algebraic extension. However, the converse is not universally true: an algebraic extension need not be finite. An algebraic extension is finite if and only if it is generated by a finite number of algebraic elements (i.e., it is a finitely generated algebraic extension). A prominent example of an infinite algebraic extension is the field of algebraic numbers $\overline{\mathbb{Q}}$ over the rational numbers $\mathbb{Q}$. While every algebraic number is the root of a polynomial with rational coefficients, the degree $[\overline{\mathbb{Q}}:\mathbb{Q}]$ is infinite because there is no finite bound on the degrees of the minimal polynomials of all algebraic numbers.

Minimal Polynomials and Degrees

For any algebraic element $\alpha$ over $K$, the set of all polynomials in $K[x]$ that evaluate to zero at $\alpha$ forms a non-zero ideal in the principal ideal domain $K[x]$. The unique monic polynomial that generates this ideal is called the minimal polynomial of $\alpha$ over $K$, often denoted as $m_{\alpha, K}(x)$. The minimal polynomial is irreducible over $K$, and any other polynomial in $K[x]$ having $\alpha$ as a root must be a multiple of the minimal polynomial.

The simple extension $K(\alpha)$ generated by a single algebraic element $\alpha$ is always a finite extension. The degree of this extension, $[K(\alpha):K]$, is exactly equal to the degree of the minimal polynomial $m_{\alpha, K}(x)$. Furthermore, the field $K(\alpha)$ is isomorphic to the quotient ring $K[x]/\langle m_{\alpha, K}(x) \rangle$, which provides a constructive way to build algebraic extensions.

Algebraic Closure

An algebraic closure of a field $K$ is an algebraic extension $\overline{K}/K$ such that the field $\overline{K}$ is algebraically closed. A field is algebraically closed if every non-constant polynomial with coefficients in that field has at least one root within the field.

Every field possesses an algebraic closure, and this closure is unique up to an isomorphism that fixes the base field $K$ pointwise. The algebraic closure can be viewed as the "maximal" algebraic extension of $K$, because any algebraic extension of $K$ can be embedded into $\overline{K}$. For instance, the Fundamental Theorem of Algebra states that the field of complex numbers $\mathbb{C}$ is the algebraic closure of the real numbers $\mathbb{R}$.

Properties and Structural Theorems

Algebraic extensions exhibit several robust structural properties that are essential for advanced algebraic constructions:

• Transitivity: If $M/L$ and $L/K$ are both algebraic extensions, then the composite extension $M/K$ is also an algebraic extension.
• Intermediate Fields: If $L/K$ is an algebraic extension and $E$ is an intermediate field (such that $K \subseteq E \subseteq L$), then $E/K$ is an algebraic extension, and $L/E$ is also an algebraic extension.
• Compositum: The compositum (the smallest field containing all of them) of any arbitrary family of algebraic extensions of $K$, taken within a common overfield, is again an algebraic extension of $K$.
• Algebraic Elements Form a Field: The set of all elements in an extension $L/K$ that are algebraic over $K$ forms a field itself. This field is called the algebraic closure of $K$ in $L$, and it is the maximal algebraic extension of $K$ contained within $L$.

Prominent Examples

• Complex over Real Numbers: The extension $\mathbb{C}/\mathbb{R}$ is a finite algebraic extension of degree 2. Every complex number $a+bi$ is a root of a quadratic polynomial with real coefficients.
• Algebraic Numbers over Rationals: The extension $\overline{\mathbb{Q}}/\mathbb{Q}$ is an infinite algebraic extension. It contains all roots of all non-zero polynomials with rational coefficients.
• Finite Fields: For any prime power $q$ and positive integer $n$, the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ is a finite algebraic extension of degree $n$. The algebraic closure of a finite field $\mathbb{F}_q$ is the union of all finite fields $\mathbb{F}_{q^n}$ for $n \ge 1$, forming an infinite algebraic extension.
• Adjoining Roots: The extension $\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}$ is a finite algebraic extension of degree 4, constructed by successively adjoining the roots of the irreducible polynomials $x^2-2$ and $x^2-3$ to the rational numbers.