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Algebraic element

In field theory, an algebraic element of a K/F is an element \alpha \in K that is a root of some non-zero with coefficients in the base F. This contrasts with transcendental elements, which are not roots of any such polynomial. Every element of the base F is algebraic over itself, as it satisfies the linear polynomial x - \beta for \beta \in F. The algebraic elements over F in an extension are closed under , subtraction, multiplication, and (non-zero) division, forming a subfield of K. A key property is that if \alpha is algebraic over F, then the F(\alpha) is finite-dimensional as a over F, with dimension equal to the degree of the minimal polynomial of \alpha over F. An extension K/F is called algebraic if every element of K is algebraic over F; such extensions include all finite extensions and are transitive, meaning if K/E and E/F are algebraic, then K/F is algebraic. Algebraic elements play a central role in , where algebraic numbers are those algebraic over \mathbb{Q}, and in , where they underpin the study of symmetries of polynomial roots. For instance, the i is algebraic over both \mathbb{R} (via x^2 + 1 = 0) and \mathbb{Q}, generating the algebraic extensions \mathbb{C}/\mathbb{R} and \mathbb{Q}(i)/\mathbb{Q}.

Definition and Context

Formal Definition

In the context of field theory, consider a base field K and an extension field L containing K. An element \alpha \in L is algebraic over K if there exists a non-zero f(x) \in K such that f(\alpha) = 0. This condition means that \alpha is a of f(x), establishing an algebraic dependence relation between \alpha and the elements of K, with the coefficients of f(x) drawn exclusively from K. If no such non-zero exists, then \alpha is transcendental over K. Algebraic numbers form a particular instance of this notion, where the base field is the rationals \mathbb{Q}; thus, an algebraic number is any complex number algebraic over \mathbb{Q}.

Role in Field Extensions

Algebraic elements play a fundamental role in the construction and study of field extensions, particularly those that are algebraic in nature. If \alpha is an algebraic element over a field K, then the simple extension K(\alpha) is an algebraic extension of K, meaning that every element in K(\alpha) is itself algebraic over K. This follows from the fact that elements of K(\alpha) can be expressed as polynomials in \alpha with coefficients in K, and since \alpha satisfies a polynomial equation over K, linear combinations and products involving \alpha also satisfy such equations. The extension K(\alpha)/K is finite-dimensional as a over K, with the dimension equal to the of the minimal polynomial of \alpha over K. This finite dimensionality underscores the structured nature of such extensions, where a basis can be taken as \{1, \alpha, \alpha^2, \dots, \alpha^{n-1}\}, with n being that . Moreover, K(\alpha) is the smallest containing both K and \alpha, serving as the quotient field of the K[\alpha]. In the broader context of field theory, every field K admits an algebraic closure, which is an algebraic extension \overline{K} of K that is algebraically closed (every non-constant polynomial with coefficients in \overline{K} has a root in \overline{K}). The existence of such a closure was established by Steinitz in 1910, ensuring that the algebraic elements over K can be comprehensively incorporated into a single extension field. This structure provides a universal setting for studying polynomials over K, as every nonconstant polynomial in K splits completely in \overline{K}.

Examples

Simple Radical Examples

One of the simplest examples of an algebraic element over the rational numbers \mathbb{Q} is the square root of 2, denoted \sqrt{2}. This element satisfies the polynomial equation x^2 - 2 = 0, which has coefficients in \mathbb{Q}. To verify, substitute \alpha = \sqrt{2} into the polynomial: \alpha^2 - 2 = 2 - 2 = 0. The polynomial f(x) = x^2 - 2 \in \mathbb{Q} is irreducible over \mathbb{Q} by Eisenstein's criterion with prime p = 2, as 2 divides the constant term -2 but not the leading coefficient 1, and $2^2 = 4 does not divide -2. Thus, \sqrt{2} is algebraic over \mathbb{Q} of degree 2, and x^2 - 2 is its monic minimal polynomial. Similarly, the of 3, denoted \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{3}, is algebraic over \mathbb{Q} as it satisfies x^3 - 3 = 0. Substituting \beta = \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{3} yields \beta^3 - 3 = 3 - 3 = 0. The g(x) = x^3 - 3 \in \mathbb{Q} is irreducible over \mathbb{Q} by with prime p = 3, since 3 divides -3 but not 1, and $9 does not divide -3. Hence, \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{3} has degree 3 over \mathbb{Q}, with x^3 - 3 as its monic minimal polynomial. Nested radicals provide another accessible example, such as \gamma = \sqrt{2 + \sqrt{3}}. To show it is algebraic over \mathbb{Q}, compute \gamma^2 = 2 + \sqrt{3}, so \gamma^2 - 2 = \sqrt{3}. Squaring both sides gives (\gamma^2 - 2)^2 = 3, which expands to \gamma^4 - 4\gamma^2 + 4 = 3, or \gamma^4 - 4\gamma^2 + 1 = 0. Thus, \gamma satisfies the quartic h(x) = x^4 - 4x^2 + 1 \in \mathbb{Q}. This is irreducible over \mathbb{Q}, as it has no rational (possible candidates \pm 1 do not satisfy it) and does not factor into quadratics with rational coefficients (assuming such a leads to contradictions in the coefficients, such as requiring square of non-squares). Moreover, \mathbb{Q}(\gamma) = \mathbb{Q}(\sqrt{2}, \sqrt{3}), which has 4 over \mathbb{Q}, confirming that h(x) is the monic minimal of 4. Primitive nth roots of unity offer further radical examples in the complex numbers. A nth root of unity \zeta = e^{2\pi i / n} satisfies the nth \Phi_n(x) = 0 over \mathbb{Q}, which is monic and irreducible. For instance, when n=3, \zeta = e^{2\pi i / 3} satisfies \Phi_3(x) = x^2 + x + 1 = 0. These polynomials define algebraic elements of degree \phi(n) over \mathbb{Q}, where \phi is .

Examples from Polynomials

Algebraic elements arise as roots of irreducible polynomials over a base field, providing concrete illustrations beyond radical expressions. Consider the quadratic polynomial x^2 + x + 1 = 0 over the rationals \mathbb{Q}. Its roots are the primitive cube roots of unity, \omega = e^{2\pi i / 3} and \omega^2, which satisfy \omega^3 = 1 and \omega \neq 1, excluding the trivial root 1. These elements generate the cyclotomic field extension \mathbb{Q}(\omega)/\mathbb{Q} of degree 2. Another quadratic example is the golden ratio \phi = \frac{1 + \sqrt{5}}{2}, which satisfies the equation x^2 - x - 1 = 0 over \mathbb{Q}. This irrational number, approximately 1.618, appears in various geometric and combinatorial contexts and adjoins to \mathbb{Q} to form a quadratic extension of degree 2. Algebraic elements also exist in finite fields. For instance, the finite field \mathrm{GF}(8) can be constructed as the quotient \mathrm{GF}(2) / (x^3 + x + 1), where x^3 + x + 1 is irreducible over \mathrm{GF}(2). Let \alpha denote the image of x in this quotient; then \alpha is an algebraic element satisfying \alpha^3 + \alpha + 1 = 0, and the elements of \mathrm{GF}(8) are \{0, 1, \alpha, \alpha+1, \alpha^2, \alpha^2+1, \alpha^2+\alpha, \alpha^2+\alpha+1\}, forming a degree-3 extension over \mathrm{GF}(2). An algebraic element \alpha over a K may satisfy multiple in K, such as any multiple of its minimal polynomial. The set of all such annihilating forms a in K, generated by the unique monic minimal polynomial of \alpha. These examples, including the above, typically generate simple extensions K(\alpha)/K.

Core Properties

Minimal Polynomial

In field theory, for an algebraic element \alpha over a K, the minimal polynomial m_\alpha(x) is defined as the monic in K of least degree such that m_\alpha(\alpha) = 0; this polynomial is unique and irreducible over K. Key properties of the minimal polynomial include that it divides any other f(x) \in K for which f(\alpha) = 0, and the K/(m_\alpha(x)) is isomorphic to the field extension K(\alpha) as fields. The existence of the minimal polynomial is ensured by the structure of rings: the of the evaluation homomorphism \phi: K \to K(\alpha) given by \phi(g(x)) = g(\alpha) is a in the K, generated by the m_\alpha(x) of minimal degree. For example, consider \alpha = \sqrt{2} over \mathbb{Q}; its minimal polynomial is m_\alpha(x) = x^2 - 2, which is irreducible over \mathbb{Q} by with prime 2. The degree of \alpha over K equals \deg(m_\alpha).

Algebraic Degree and Conjugates

The degree of an algebraic element \alpha over a K, denoted \deg(\alpha) or simply the degree of \alpha, is defined as the degree of its minimal polynomial m_\alpha(x) over K. This degree equals the degree of the field extension [K(\alpha):K], which is the of K(\alpha) as a over K. A standard basis for this vector space consists of the powers \{1, \alpha, \alpha^2, \dots, \alpha^{n-1}\}, where n = \deg(m_\alpha(x)). The conjugates of \alpha over K are the roots of the minimal polynomial m_\alpha(x) in an algebraic closure \overline{K} of K. Equivalently, these conjugates are the images \sigma(\alpha), where \sigma ranges over all K-embeddings of K(\alpha) into \overline{K}. There are exactly n such conjugates, counting multiplicities, and each conjugate is algebraic over K with the same minimal polynomial and degree as \alpha. For example, the golden ratio \phi = \frac{1 + \sqrt{5}}{2} has minimal polynomial x^2 - x - 1 over \mathbb{Q}, so its degree is 2 and [\mathbb{Q}(\phi):\mathbb{Q}] = 2. The conjugates of \phi are \phi itself and \frac{1 - \sqrt{5}}{2}, the two roots of this polynomial.

Advanced Characteristics

Trace and Norm

In the context of a finite field extension K(\alpha)/K where \alpha is an algebraic element over the base field K, the trace and norm are fundamental K-linear and multiplicative maps, respectively, that serve as field invariants capturing symmetric properties of \alpha and its conjugates. The trace \operatorname{Tr}_{K(\alpha)/K}(\alpha) is defined as the sum of the conjugates of \alpha, where the conjugates are the roots \alpha_1 = \alpha, \alpha_2, \dots, \alpha_n of the minimal m_\alpha(x) of \alpha over K, with n = [K(\alpha):K] the of the extension. Equivalently, with respect to the power basis \{1, \alpha, \dots, \alpha^{n-1}\}, the is the of the K- of multiplication by \alpha on K(\alpha). The norm N_{K(\alpha)/K}(\alpha) is the product of these conjugates, \prod_{i=1}^n \alpha_i. It coincides with the of the multiplication-by-\alpha map on the same basis. These quantities can be computed directly from the minimal polynomial m_\alpha(x) = x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \in K, via : the trace is \operatorname{Tr}_{K(\alpha)/K}(\alpha) = -a_{n-1}, and the norm is N_{K(\alpha)/K}(\alpha) = (-1)^n a_0. The norm satisfies the multiplicative property: for any \beta, \gamma \in K(\alpha), N_{K(\alpha)/K}(\beta \gamma) = N_{K(\alpha)/K}(\beta) \cdot N_{K(\alpha)/K}(\gamma).

Powers and Sums of Algebraic Elements

Algebraic elements over a base K exhibit closure properties under fundamental field operations, ensuring that the collection of such elements forms a subfield algebraic over K. Specifically, if \alpha and \beta are algebraic over K, then their sum \alpha + \beta and product \alpha \beta are also algebraic over K. This follows from the fact that both \alpha + \beta and \alpha \beta lie in the simple extension K(\alpha, \beta), which is a finite-degree extension of K since [K(\alpha, \beta):K] \leq [K(\alpha):K] \cdot [K(\beta):K] < \infty. Similarly, any power \alpha^k for integer k \geq 1 remains algebraic over K, as it resides in the finite extension K(\alpha). In the context of a simple algebraic extension K(\alpha), where \alpha has minimal polynomial of degree n over K, every element can be uniquely represented as a \sum_{i=0}^{n-1} c_i \alpha^i with coefficients c_i \in K. This set \{1, \alpha, \dots, \alpha^{n-1}\} forms a basis for K(\alpha) as a over K. Multiplication by \alpha in this extension shifts the basis elements and reduces higher powers using the relation from the minimal polynomial m_\alpha(\alpha) = 0, ensuring under within the basis representation. The powers of an algebraic element \alpha satisfy a linear derived from its minimal . If m_\alpha(X) = X^n + a_{n-1} X^{n-1} + \dots + a_0, then for k \geq n, \alpha^k = -a_{n-1} \alpha^{k-1} - \dots - a_0 \alpha^{k-n}. This recurrence, analogous to the Cayley-Hamilton theorem for the of the minimal , allows efficient computation of higher powers by reducing them to lower-degree terms in the basis. For a concrete illustration, consider \alpha = \sqrt{2}, which satisfies the minimal polynomial X^2 - 2 = 0 over \mathbb{Q}. Then \alpha^2 = 2, and higher powers reduce accordingly: \alpha^3 = \alpha \cdot \alpha^2 = 2\alpha, \alpha^4 = 2 \alpha^2 = 4, and so on, all expressible in the basis \{1, \alpha\}. This demonstrates the practical reduction of powers in the extension \mathbb{Q}(\alpha).

Related Concepts

Algebraic Integers

An is a \alpha that is algebraic over \mathbb{Q} and satisfies a equation with coefficients in \mathbb{Z}. This condition ensures that \alpha is integral over \mathbb{Z}, meaning it behaves like an in the ring-theoretic sense within its field of definition. Equivalently, the minimal of \alpha over \mathbb{Q} is and has coefficients in \mathbb{Z}. A classic example is \sqrt{2}, whose minimal polynomial is x^2 - 2 = 0, which is monic with integer coefficients, making \sqrt{2} an algebraic integer. In contrast, $1/\sqrt{2} is algebraic over \mathbb{Q} with minimal polynomial x^2 - 1/2 = 0, but this polynomial has a non-integer coefficient, so $1/\sqrt{2} is not an algebraic integer; its monic minimal polynomial over \mathbb{Q} fails to have integer coefficients. For an algebraic integer \alpha of degree n over \mathbb{Q}, the ring \mathbb{Z}[\alpha] = \{ a_0 + a_1 \alpha + \cdots + a_{n-1} \alpha^{n-1} \mid a_i \in \mathbb{Z} \} consists of all integer linear combinations of powers of \alpha and forms a subring of the algebraic integers. The full ring of algebraic integers is the integral closure of \mathbb{Z} in \mathbb{C}, comprising all complex numbers integral over \mathbb{Z}. The of the \mathbb{Z}[\alpha] is defined using the power basis \{1, \alpha, \dots, \alpha^{n-1}\} and equals the of the minimal of \alpha, which relates to (and often divides) the of the number field \mathbb{Q}(\alpha). This quantity measures the "ramification" or arithmetic complexity of the extension without delving into explicit calculations.

Transcendental Elements

In extensions, an β in an extension L of a base K is defined as transcendental over K if it is not algebraic over K, meaning no non-zero with coefficients in K has β as a . This contrasts with algebraic elements, which generate finite-degree extensions, whereas a transcendental β produces a K(β)/K of infinite degree [K(β):K] = ∞, lacking a finite basis as a over K. The simple transcendental extension K(β)/K is isomorphic to the field of rational functions K(x), where x is an indeterminate, highlighting its structural similarity to function fields. Prominent examples include the mathematical constants π and , both transcendental over the of rational numbers ℚ. Another standard example arises in function fields, where the indeterminate x serves as a transcendental element over the base K. A foundational result establishing specific instances of transcendence is the , which asserts that e^a is transcendental over ℚ for any non-zero a. This theorem directly implies the transcendence of e, obtained by setting a = 1, and of π, since e^{iπ} = -1 is algebraic over ℚ while i is algebraic, so iπ algebraic would contradict the theorem.

Applications

In Galois Theory

In , the of an extension K(\alpha)/K, where \alpha is algebraic over K, plays a central role in understanding the structure of the extension through its action on the roots of the minimal m_\alpha(x) of \alpha. Specifically, if L/K is a containing \alpha, then the \mathrm{Gal}(L/K) acts on \alpha by sending it to other roots of m_\alpha(x), and these images are known as the Galois conjugates of \alpha. For the K(\alpha)/K, if it is separable, the \mathrm{Gal}(K(\alpha)/K) permutes the roots of m_\alpha(x) transitively, and its order equals the degree [K(\alpha):K]. An algebraic extension K(\alpha)/K is normal if and only if the minimal polynomial m_\alpha(x) splits completely into linear factors over K(\alpha), meaning all roots of m_\alpha(x) lie in K(\alpha). In this case, K(\alpha)/K is the of m_\alpha(x) over K, and the extension is . If m_\alpha(x) does not split completely in K(\alpha), the full of m_\alpha(x) over K provides the smallest extension containing \alpha, with the acting faithfully on the roots. The concept of solvability by radicals connects algebraic elements to the solvability of their defining equations. An algebraic element \alpha over K (of characteristic zero) is solvable by radicals if the splitting field of its minimal m_\alpha(x) over K can be obtained by a tower of radical extensions, which occurs precisely when the of that is a . This criterion, established by Galois, implies that polynomials with solvable s, such as quadratics or cubics, admit solutions expressible in radicals, while those with nonsolvable groups, like the general quintic, do not. A concrete example arises with roots of unity, which generate cyclotomic extensions. The cube roots of unity, roots of the 3rd cyclotomic polynomial \Phi_3(x) = x^2 + x + 1, adjoin to \mathbb{Q} to form the extension \mathbb{Q}(\zeta_3)/\mathbb{Q}, where \zeta_3 = e^{2\pi i / 3}. This is a Galois extension with Galois group isomorphic to (\mathbb{Z}/3\mathbb{Z})^\times \cong \mathbb{Z}/2\mathbb{Z}, which is abelian and hence solvable, confirming that the roots are solvable by radicals. More generally, the nth cyclotomic extension \mathbb{Q}(\zeta_n)/\mathbb{Q} has Galois group (\mathbb{Z}/n\mathbb{Z})^\times, always abelian and thus solvable.

In Algebraic Number Theory

In algebraic number theory, algebraic elements play a central role in the study of number fields, which are finite field extensions K/\mathbb{Q}. Such a field K can often be expressed as K = \mathbb{Q}(\alpha) where \alpha is an algebraic integer, and the degree of the extension is n = [K : \mathbb{Q}], equal to the degree of the minimal polynomial of \alpha over \mathbb{Q}. This construction allows the arithmetic properties of K to be analyzed through the adjunction of a single algebraic element, facilitating the exploration of primes, units, and ideals within K. The \mathcal{O}_K of a number field K is defined as the integral closure of \mathbb{Z} in K, comprising all elements of K that are over \mathbb{Z}. For quadratic fields K = \mathbb{Q}(\sqrt{d}) where d is a , \mathcal{O}_K = \mathbb{Z}[\sqrt{d}] when d \equiv 2 or $3 \pmod{4}, providing a \mathbb{Z}-basis for the ring. This ring \mathcal{O}_K is a Dedekind domain, meaning it is Noetherian, integrally closed in K, and every nonzero ideal factors uniquely into prime ideals. In this setting, the norm of a prime ideal \mathfrak{p} of \mathcal{O}_K lying above a rational prime p, denoted N(\mathfrak{p}), extends the norm of algebraic elements by N(\mathfrak{p}) = p^f where f is the residue degree, enabling the study of how rational primes decompose, split, or ramify in \mathcal{O}_K. The class number h_K of a number field K is the order of the \mathrm{Cl}(\mathcal{O}_K), which measures the extent to which unique factorization fails for elements in \mathcal{O}_K, as opposed to the unique factorization of ideals guaranteed in Dedekind domains. This finiteness of h_K follows from , and computations often involve the , a derived from the unit group \mathcal{O}_K^\times via , which states that \mathcal{O}_K^\times \cong \mu(K) \times \mathbb{Z}^{r_1 + r_2 - 1} where \mu(K) is the group of roots of unity in K, r_1 is the number of real embeddings, and r_2 is half the number of complex embeddings. For instance, in the \mathbb{Q}(\sqrt{-5}), the prime 2 ramifies as (2) = \mathfrak{p}^2 where \mathfrak{p} = (2, 1 + \sqrt{-5}) is a non-principal ideal, contributing to the class number h_K = 2 and illustrating the arithmetic obstructions to unique element factorization.