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Algebraic number

An algebraic number is a that is a of a non-zero with rational coefficients. Equivalently, it satisfies a polynomial equation with coefficients after clearing denominators, and the minimal such over is unique and monic. Examples include all rational numbers, as roots of linear polynomials, and numbers like √2, which is a of x² - 2 = 0. The set of all algebraic numbers, often denoted 𝔸, forms a subfield of the numbers that is algebraic over ℚ, meaning every element is algebraic over ℚ. This field is countable, despite being dense in the , and it is the of ℚ within ℂ. In contrast, transcendental numbers, such as π and e, are not roots of any such and constitute almost all numbers in the sense of . Algebraic numbers are central to algebraic number theory, which extends arithmetic properties of the integers to more general settings. A key subset consists of the algebraic integers, which are algebraic numbers that are roots of monic polynomials with integer coefficients; for instance, √2 is an algebraic integer, while 1/√2 is algebraic but not an integral. The algebraic integers within a finite extension K of ℚ, called a number field, form the ring of integers 𝒪_K, which is a Dedekind domain and supports unique factorization into prime ideals. Number fields, such as the quadratic field ℚ(√d) for square-free integer d, enable the study of generalizations of Diophantine equations and class field theory.

Fundamentals

Definition

An algebraic number is a \alpha that is a of some non-zero P(x) with rational coefficients, meaning there exist rational numbers a_n, \dots, a_0, not all zero, such that P(\alpha) = a_n \alpha^n + \dots + a_1 \alpha + a_0 = 0. Equivalently, \alpha is algebraic over \mathbb{Q} if it belongs to some finite-degree of the field of rational numbers \mathbb{Q}. Numbers that are not algebraic in this sense are called transcendental. The concept of algebraic numbers was formalized through the foundational work of Carl Friedrich Gauss and other mathematicians in the 19th century, particularly in the context of number theory and field extensions. Gauss's Disquisitiones Arithmeticae (1801) played a pivotal role in systematizing the arithmetic of such numbers. The collection of all algebraic numbers constitutes a field, commonly denoted \overline{\mathbb{Q}}, which is the algebraic closure of \mathbb{Q}; this structure encompasses all roots of rational polynomials and supports the operations of addition, subtraction, multiplication, and division (except by zero).

Examples

All rational numbers are algebraic, as each q \in \mathbb{Q} satisfies the monic linear polynomial equation x - q = 0 with rational coefficients. Quadratic irrationals illustrate irrational algebraic numbers of degree 2. The number \sqrt{2} is algebraic, being a root of the polynomial x^2 - 2 = 0. Similarly, the golden ratio \phi = \frac{1 + \sqrt{5}}{2} is algebraic as a root of x^2 - x - 1 = 0. Examples of algebraic numbers of higher degree include the real of 2, a root of x^3 - 2 = 0. Another is a real root of the irreducible cubic x^3 - x - 1 = 0, which has degree 3 over \mathbb{Q}. Complex algebraic numbers abound, such as i, the , which is a root of x^2 + 1 = 0. Primitive nth roots of unity, for instance e^{2\pi i / 3}, are algebraic, satisfying the nth cyclotomic over \mathbb{Q}. Nested radicals often produce algebraic numbers. The infinite nested radical \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}} converges to 2, a rational algebraic number solving x^2 - x - 2 = 0.

Properties

Minimal Polynomials

For an algebraic number \alpha, the minimal polynomial m_\alpha(x) is defined as the monic irreducible polynomial of least degree in \mathbb{Q} such that m_\alpha(\alpha) = 0. This polynomial is unique, and it generates the ideal of all polynomials in \mathbb{Q} that vanish at \alpha. Moreover, the field extension \mathbb{Q}(\alpha) is isomorphic to the quotient ring \mathbb{Q} / (m_\alpha(x)). A of the minimal is that it divides any other in \mathbb{Q} that has \alpha as a root. The roots of m_\alpha(x) are precisely the conjugates of \alpha, which are the images of \alpha under embeddings of \mathbb{Q}(\alpha) into \mathbb{C} that fix \mathbb{Q}. conjugates of \alpha share the same minimal . For example, consider \alpha = \sqrt{2}. Its minimal polynomial is m_\alpha(x) = x^2 - 2, which is monic, irreducible over \mathbb{Q}, and satisfies m_\alpha(\sqrt{2}) = 0. The roots are \sqrt{2} and -\sqrt{2}, the conjugates of \sqrt{2}. Another example is the \phi = \frac{1 + \sqrt{5}}{2}, whose minimal polynomial is m_\phi(x) = x^2 - x - 1. This polynomial is irreducible over \mathbb{Q} by the , and its roots are \phi and \frac{1 - \sqrt{5}}{2}.

Degree and Field Extensions

The degree of an algebraic number \alpha, denoted \deg(\alpha), is defined as the degree of its minimal polynomial m_\alpha(x) over \mathbb{Q}. This degree provides a measure of the "complexity" of \alpha as a root of a polynomial with rational coefficients. For an algebraic number \alpha of n = \deg(\alpha), the \mathbb{Q}(\alpha) forms a simple extension of \mathbb{Q} with [\mathbb{Q}(\alpha) : \mathbb{Q}] = n. In this extension, the set \{1, \alpha, \alpha^2, \dots, \alpha^{n-1}\} serves as a basis for \mathbb{Q}(\alpha) as a vector space over \mathbb{Q}. Consequently, every element \beta \in \mathbb{Q}(\alpha) can be uniquely expressed as \beta = a_0 + a_1 \alpha + \dots + a_{n-1} \alpha^{n-1} where a_i \in \mathbb{Q}. An element \alpha \in \mathbb{C} is algebraic over \mathbb{Q} if and only if the extension \mathbb{Q}(\alpha)/\mathbb{Q} has finite degree. This criterion links the algebraic nature of numbers directly to the dimensionality of their generated field extensions. In more general settings, the tower law states that for fields L \supseteq K \supseteq F, the degree [L : F] = [L : K] \cdot [K : F], which applies to composite extensions involving algebraic numbers.

Algebraic versus Transcendental Numbers

A is defined as a that is not algebraic, meaning it is not a of any non-zero with rational coefficients. The existence of transcendental numbers was established in 1844 by through his approximation theorem, which demonstrated that certain constructed numbers, known as Liouville numbers, cannot be roots of any such polynomial and are thus transcendental. Building on this foundation, Charles Hermite proved in 1873 that the base of the natural logarithm, e, is transcendental, marking the first such proof for a specific non-constructed constant. extended these ideas in 1882 by proving that \pi is transcendental, resolving the question of the squaring of the circle in . Transcendental numbers are more numerous than algebraic numbers in the , as the latter form a while the former have the . The algebraic numbers constitute a of the finite sets of roots of all polynomials with rational coefficients, ensuring their countability. Despite their countability, the algebraic numbers are dense in \mathbb{C}.

Number Fields

Simple Algebraic Extensions

A simple algebraic extension of the rational numbers \mathbb{Q} is a field extension K/\mathbb{Q} generated by a single algebraic number \alpha, denoted K = \mathbb{Q}(\alpha). By the , every finite extension of \mathbb{Q} is simple, as extensions of fields of characteristic zero are separable. The degree [K : \mathbb{Q}] equals the degree of the minimal polynomial m_\alpha(x) of \alpha over \mathbb{Q}. In such an extension, the \mathbb{Q}-embeddings of K into \mathbb{C} are determined by sending \alpha to one of the roots of m_\alpha(x). If K/\mathbb{Q} is Galois (hence normal and separable), the \mathrm{Gal}(K/\mathbb{Q}) acts faithfully on K by \mathbb{Q}-automorphisms, permuting the roots of m_\alpha(x) transitively. More precisely, each \sigma \in \mathrm{Gal}(K/\mathbb{Q}) is determined by \sigma(\alpha), which is a root of m_\alpha(x), and the action preserves the field structure since all roots lie in K. The extension K/\mathbb{Q} is normal if and only if m_\alpha(x) splits completely into linear factors over K. The discriminant of K, denoted \mathrm{disc}(K), measures the "size" of the extension and arises from the embeddings \sigma_i : K \hookrightarrow \mathbb{C} (for i = 1, \dots, n where n = [K : \mathbb{Q}]). It is given by \mathrm{disc}(K) = \prod_{1 \leq i < j \leq n} (\sigma_i(\alpha) - \sigma_j(\alpha))^2, up to a sign depending on the parity of n(n-1)/2; explicit computation of this quantity is typically deferred to specific cases using the norm of the derivative of m_\alpha(x). A concrete example is K = \mathbb{Q}(\sqrt{2}), where \alpha = \sqrt{2} has minimal polynomial m_\alpha(x) = x^2 - 2, so [K : \mathbb{Q}] = 2. This extension is Galois with \mathrm{Gal}(K/\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z}, generated by the automorphism \sigma: \sqrt{2} \mapsto -\sqrt{2}, which permutes the roots \{\sqrt{2}, -\sqrt{2}\} of m_\alpha(x).

Algebraic Closure

The algebraic closure of the rational numbers \mathbb{Q}, denoted \overline{\mathbb{Q}}, is defined as the union of all finite field extensions of \mathbb{Q} within the complex numbers \mathbb{C}. This construction yields the smallest algebraically closed field containing \mathbb{Q}, consisting precisely of all numbers algebraic over \mathbb{Q}. A key property of \overline{\mathbb{Q}} is that it is algebraically closed: every non-constant polynomial with coefficients in \overline{\mathbb{Q}} factors completely into linear factors over \overline{\mathbb{Q}}. The field \overline{\mathbb{Q}} is countable, as it arises from the countable union of the roots of countably many polynomials with integer coefficients, each having finitely many roots. Since all elements of \overline{\mathbb{Q}} are algebraic over \mathbb{Q}, its transcendence degree over \mathbb{Q} is zero. Every element of \overline{\mathbb{Q}} embeds into \mathbb{C}, as the fundamental theorem of algebra ensures that polynomials over \mathbb{Q} split in \mathbb{C}, allowing the realization of \overline{\mathbb{Q}} as a subfield of \mathbb{C}. Consequently, the intersection \overline{\mathbb{Q}} \cap \mathbb{R} forms the field of real algebraic numbers, comprising all real roots of polynomials with rational coefficients. The absolute Galois group \mathrm{Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) is profinite, arising as the inverse limit of the Galois groups of all finite Galois extensions of \mathbb{Q}.

Algebraic Integers

Definition and Basic Properties

An algebraic integer is defined as a complex number \alpha that is algebraic over \mathbb{Q} and whose minimal polynomial m_\alpha(x) over \mathbb{Q} is monic with integer coefficients, meaning m_\alpha(x) \in \mathbb{Z}. Equivalently, \alpha is an algebraic integer if it is a root of some monic polynomial with coefficients in \mathbb{Z}. This definition distinguishes algebraic integers from general algebraic numbers, which may have minimal polynomials with rational but non-integer coefficients. All ordinary integers in \mathbb{Z} are algebraic integers, as each n \in \mathbb{Z} satisfies the monic polynomial x - n = 0 with integer coefficients. For example, \sqrt{2} is an algebraic integer because its minimal polynomial is x^2 - 2 = 0 \in \mathbb{Z}, whereas $1/\sqrt{2} is not, since its minimal polynomial x^2 - 1/2 = 0 has a non-integer coefficient. The set of all algebraic integers, denoted \overline{\mathbb{Z}}, forms a subring of the complex numbers under addition and multiplication. Specifically, the sum and product of any two algebraic integers are themselves algebraic integers, ensuring closure under ring operations. A fundamental theorem states that an algebraic number \alpha is an algebraic integer if and only if its minimal polynomial over \mathbb{Q} has integer coefficients. In the context of a number field K = \mathbb{Q}(\alpha) where \alpha is an algebraic integer of degree n = [K : \mathbb{Q}], the elements \{1, \alpha, \dots, \alpha^{n-1}\} may form an integral basis for the ring of integers of K, meaning \overline{\mathbb{Z}} \cap K = \mathbb{Z}[\alpha] with this power basis generating the ring as a \mathbb{Z}-module.

Rings of Algebraic Integers

In a number field K, the ring of integers \mathcal{O}_K is defined as the integral closure of \mathbb{Z} in K, consisting of all elements in K that satisfy monic polynomials with integer coefficients. This construction ensures that \mathcal{O}_K captures the "integral" elements within the field, forming a subring that extends the rational integers. The ring \mathcal{O}_K possesses several key structural properties. It is a Dedekind domain, meaning it is an integrally closed Noetherian domain in which every nonzero prime ideal is maximal. Additionally, \mathcal{O}_K is finitely generated as a \mathbb{Z}-module with rank equal to the degree [K:\mathbb{Q}], allowing it to be expressed with respect to an integral basis. These features underpin much of the arithmetic theory in number fields. The discriminant of \mathcal{O}_K, denoted \operatorname{disc}(\mathcal{O}_K), is a fundamental invariant that quantifies the ramification of prime ideals from \mathbb{Q} to K. It is computed as the determinant of the trace form matrix with respect to an integral basis and plays a crucial role in measuring how the extension distorts lattice structures. For quadratic fields K = \mathbb{Q}(\sqrt{d}) where d is a square-free integer not congruent to 1 modulo 4, the discriminant is $4d; otherwise, it is d. Regarding units and ideals, the unit group \mathcal{O}_K^\times of \mathcal{O}_K is finitely generated, and Dirichlet's unit theorem describes its structure: it is isomorphic to \mathbb{Z}^{r_1 + r_2 - 1} \times \mu_K, where r_1 is the number of real embeddings, $2r_2 is the number of complex embeddings, and \mu_K is the torsion subgroup of roots of unity in K. Ideals in \mathcal{O}_K factor uniquely into prime ideals, reflecting its Dedekind nature. A concrete example illustrates these concepts: for the quadratic field K = \mathbb{Q}(\sqrt{5}), the ring of integers is \mathcal{O}_K = \mathbb{Z}\left[\frac{1 + \sqrt{5}}{2}\right], which has discriminant 5 and unit group generated by the fundamental unit \frac{1 + \sqrt{5}}{2}.

Special Classes

Quadratic Algebraic Numbers

Quadratic algebraic numbers are elements of quadratic fields, which are field extensions of the rational numbers \mathbb{Q} of degree 2. Specifically, for a square-free integer d \neq 0, 1, the quadratic field is K = \mathbb{Q}(\sqrt{d}), consisting of all elements of the form a + b\sqrt{d} where a, b \in \mathbb{Q}. These fields are classified as real if d > 0 or imaginary if d < 0, and every algebraic number of degree 2 over \mathbb{Q} generates such a field. Any \alpha \in K satisfies a minimal polynomial of degree 2 over \mathbb{Q}, given by x^2 - \operatorname{tr}(\alpha)x + N(\alpha) = 0, where \operatorname{tr}(\alpha) is the trace and N(\alpha) is the norm of \alpha. The trace \operatorname{tr}(a + b\sqrt{d}) = 2a is the sum of the conjugates a + b\sqrt{d} and a - b\sqrt{d}, while the norm N(a + b\sqrt{d}) = a^2 - d b^2 is their product. These maps are essential for studying arithmetic in K, as they provide a way to relate elements back to \mathbb{Q}. The ring of integers \mathcal{O}_K of K comprises the algebraic integers in K. For d \equiv 2, 3 \pmod{4}, \mathcal{O}_K = \mathbb{Z}[\sqrt{d}]; for d \equiv 1 \pmod{4}, \mathcal{O}_K = \mathbb{Z}\left[\frac{1 + \sqrt{d}}{2}\right]. The norm on \mathcal{O}_K extends that on K, and it detects failures of unique factorization; for instance, in \mathbb{Z}[\sqrt{-5}], the element 6 factors as $2 \cdot 3 and as (1 + \sqrt{-5})(1 - \sqrt{-5}), both products of irreducibles, showing it is not a unique factorization domain. In real quadratic fields (d > 0), the unit group of \mathcal{O}_K is generated by -1 and a fundamental unit \epsilon > 1, whose solutions arise from the Pell equation x^2 - d y^2 = \pm 1. The minimal positive solution (\epsilon, 1) yields \epsilon = x + y \sqrt{d}, and all units are \pm \epsilon^k for k \in \mathbb{Z}. A prominent example is the Gaussian integers, the ring of integers of \mathbb{Q}(i) where d = -1. Here, \mathcal{O}_K = \mathbb{Z} = \{a + bi \mid a, b \in \mathbb{Z}\}, equipped with the norm N(a + bi) = a^2 + b^2. This ring is a Euclidean domain under the norm, hence a principal ideal domain and unique factorization domain.

Cyclotomic Numbers

Cyclotomic numbers are the algebraic numbers that arise as roots of unity, specifically the primitive nth roots of unity \zeta_n = e^{2\pi i / n} for positive integers n, which satisfy \zeta_n^n = 1 and have minimal order n. These numbers generate the cyclotomic fields \mathbb{Q}(\zeta_n), which are finite Galois extensions of the rational numbers \mathbb{Q}. The nth cyclotomic field \mathbb{Q}(\zeta_n) is the splitting field over \mathbb{Q} of the nth cyclotomic polynomial \Phi_n(x), defined as the monic polynomial \Phi_n(x) = \prod (x - \zeta_n^k), where the product runs over all integers k between $1andnthat are coprime ton. This polynomial is irreducible over \mathbb{Q}$ and has integer coefficients. The degree of the extension [\mathbb{Q}(\zeta_n) : \mathbb{Q}] equals \varphi(n), where \varphi denotes , which counts the number of integers from $1toncoprime ton. For example, when n = pis prime,\varphi(p) = p-1and\Phi_p(x) = (x^p - 1)/(x - 1), which is irreducible over \mathbb{Q}. The ring of integers of \mathbb{Q}(\zeta_n)is\mathcal{O}_K = \mathbb{Z}[\zeta_n], the ring generated by \mathbb{Z}and\zeta_n, and this holds for all nsince\mathbb{Z}[\zeta_n]is integrally closed in\mathbb{Q}(\zeta_n)$. The \mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) is isomorphic to (\mathbb{Z}/n\mathbb{Z})^\times, the of integers n coprime to n, which is abelian. The isomorphism arises from of group elements: for k \in (\mathbb{Z}/n\mathbb{Z})^\times, the \sigma_k satisfies \sigma_k(\zeta_n) = \zeta_n^k. This abelian structure makes cyclotomic fields prototypical examples of abelian extensions of \mathbb{Q}, and by the Kronecker-Weber theorem, every abelian extension of \mathbb{Q} is contained in some . play a central role in s in , such as determining regular primes (where p does not divide the class number of \mathbb{Q}(\zeta_p)) and Weber's class number problem for real cyclotomic extensions. A key subfield of \mathbb{Q}(\zeta_n) is its maximal real subfield \mathbb{Q}(\zeta_n + \zeta_n^{-1}) = \mathbb{Q}(2 \cos(2\pi / n)), which has degree \varphi(n)/2 over \mathbb{Q} for n > 2. This subfield consists of the fixed points under complex conjugation and is generated by the algebraic integer \zeta_n + \zeta_n^{-1}.

Applications and Related Concepts

In Number Theory

Algebraic numbers play a central role in , which studies how well irrational algebraic numbers can be approximated by rational numbers. A landmark result is , which states that for any irrational algebraic number \alpha of degree at least 2 and any \varepsilon > 0, there are only finitely many rational approximations p/q (with p, q \in \mathbb{Z}, q > 0) satisfying |\alpha - p/q| < 1/q^{2+\varepsilon}. This bound sharpens earlier results by Thue and Siegel, establishing that algebraic irrationals cannot be approximated by rationals better than quadratically up to a small error term, with profound implications for the distribution of algebraic numbers among rationals. In the context of Diophantine equations, algebraic numbers underpin the solution to Fermat's Last Theorem, which asserts that there are no positive integers a, b, c, n with n > 2 satisfying a^n + b^n = c^n. proved this in 1995 (announced in 1994) by establishing the modularity of semistable elliptic curves over \mathbb{Q}, linking them to modular forms via Galois representations attached to number fields; the proof relies heavily on algebraic integers in rings of integers of these fields to construct and analyze the necessary Frey curves and their deformations. This resolution not only closes a centuries-old but also advances the understanding of elliptic curves as arithmetic objects tied to . Class field theory provides a complete description of all abelian Galois extensions of a number field K, parametrizing them by the ideal class group of the ring of algebraic integers \mathcal{O}_K. Specifically, it establishes a bijection between the abelian extensions of K and the subgroups of the idele class group, with the Artin map realizing this correspondence and enabling the explicit construction of such extensions via ray class groups. This framework, developed through contributions from Hilbert, Takagi, and Artin, reveals the deep interplay between ideals in \mathcal{O}_K and the Galois groups of abelian extensions, forming a cornerstone of modern algebraic number theory. L-functions associated to number fields generalize the and encode arithmetic data, such as the distribution of primes in ideals. For a number field K with \mathcal{O}_K, the is defined as \zeta_K(s) = \sum_{\mathfrak{a}} 1/N(\mathfrak{a})^s, where the sum runs over nonzero ideals \mathfrak{a} of \mathcal{O}_K and N(\mathfrak{a}) is the ; it admits an to the with a simple pole at s=1, and its residue relates to the class number and of K via the . These functions facilitate the study of prime factorization in extensions and underpin analytic methods in , including the proof of Dirichlet's theorem on primes in arithmetic progressions over number fields. As of 2025, the has seen landmark progress, including the 2024 proof of the geometric Langlands conjecture by a team of nine mathematicians led by Dennis Gaitsgory and Sam Raskin. This nearly 1,000-page proof establishes deep connections between Galois representations of \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) and automorphic forms on reductive groups over \mathbb{Q} using geometric methods involving sheaves on Riemann surfaces. It builds on earlier developments, such as the construction of compatible systems of Galois representations attached to cuspidal automorphic representations for unitary groups and further evidence for the functoriality principle, with Gaitsgory recognized by the 2025 for his foundational contributions. This ongoing work bridges and arithmetic geometry, promising deeper insights into the arithmetic of algebraic numbers through correspondences that generalize to non-abelian settings.

Expressibility by Radicals

An algebraic number \alpha is expressible by radicals over \mathbb{Q} if it belongs to a radical extension of \mathbb{Q}, which is a finite tower of extensions \mathbb{Q} = K_0 \subset K_1 \subset \cdots \subset K_m with \alpha \in K_m such that for each i, K_{i+1} = K_i(\sqrt[n_i]{\beta_i}) for some \beta_i \in K_i and integer n_i \geq 2. Equivalently, if \alpha is a root of an f(x) \in \mathbb{Q}, then \alpha is expressible by radicals all roots of f(x) can be so expressed, meaning f(x) is solvable by radicals. By , a f(x) \in \mathbb{Q} is solvable by radicals if and only if the of its K over \mathbb{Q} is a , i.e., it admits a with cyclic factors. For polynomials, the is cyclic of order 2, so roots are always expressible by radicals via the . Cubic polynomials are also solvable by radicals using Cardano's formula, which reduces the equation ax^3 + bx^2 + cx + d = 0 (after depressing to x^3 + px + q = 0) to roots of the form u + v where u^3 and v^3 solve a and satisfy $3uv + p = 0. For example, the real root of x^3 - 15x - 4 = 0 is \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{10 + \sqrt{69}} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{10 - \sqrt{69}}. Quartic polynomials are solvable by radicals, as their are subgroups of S_4, which has a . In contrast, the general quintic polynomial is not solvable by radicals, as established by the Abel–Ruffini theorem: there is no general formula expressing the roots of a fifth-degree in terms of radicals over \mathbb{Q}. proved this impossibility for the quintic in 1824, building on Paolo Ruffini's earlier partial result from 1799; the proof relies on showing that the S_5 is not solvable, due to the simplicity of its alternating subgroup A_5. A notable complication arises in the for irreducible cubic polynomials over \mathbb{Q} with three distinct real roots (when the is positive): Cardano's formula expresses these real roots using cube roots of complex (nonreal) numbers, even though real radicals alone are insufficient. For instance, the roots of x^3 - 3x + 1 = 0 involve cube roots of complex conjugates like \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{1 + i\sqrt{2}} and \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{1 - i\sqrt{2}}. confirms this necessity, as the (order 3, not a power of 2) prevents expression via real radicals for such irreducibles. The set of algebraic numbers expressible by radicals forms a proper subclass of the closed-form numbers, which are those in the smallest subfield of \mathbb{C} closed under addition, multiplication, division, exponentials, and logarithms starting from \mathbb{Q}. While all radical-expressible algebraics are closed-form (via roots as exponentials of logarithms), the converse requires ; notably, \pi is closed-form via \pi = -i \log(-1) despite being transcendental and not algebraic. Cyclotomic fields, generated by of , are radical extensions of \mathbb{Q}, as they adjoin nth of 1.