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Algebraically closed field

In , an algebraically closed is a F such that every non-constant with coefficients in F has at least one root in F. This property is equivalent to the condition that every non-constant in F factors completely into a product of linear factors over F. Prominent examples include the field of complex numbers \mathbb{C}, which is algebraically closed by the stating that every non-constant with complex coefficients splits into linear factors over \mathbb{C}. Another example is the field of algebraic numbers, the of the rational numbers \mathbb{Q}, consisting of all complex numbers algebraic over \mathbb{Q}. For any field K, there exists an , defined as an \overline{K} of K that is itself algebraically closed. Such an algebraic closure is unique up to as a of K. Algebraically closed fields play a central role in , particularly in , where they serve as the ambient setting for studying splitting fields and Galois groups. In , working over an algebraically closed field simplifies the study of varieties, enabling key results like , which establishes a correspondence between radical ideals in the and algebraic varieties.

Definition and Motivation

Definition

In , a F is defined to be algebraically closed if every non-constant in the F has at least one in F. Formally, for any p(x) \in F of n \geq 1, there exists \alpha \in F such that p(\alpha) = 0. This property ensures that F contains solutions to all its own equations within the field itself. An equivalent characterization is that F is algebraically closed if and only if every non-constant polynomial in F factors completely into linear factors over F. As a direct consequence, the only irreducible polynomials in F are the linear ones (of degree 1), up to multiplication by units in F. Since the polynomial ring F over any field F is a Euclidean domain—and hence a unique factorization domain—this implies that every element of F admits a unique factorization into irreducible elements, all of which are linear polynomials in the algebraically closed case. Thus, non-constant polynomials factor uniquely (up to units and ordering of factors) as products of linear terms, reflecting the complete solvability of polynomial equations over F.

Historical Motivation

The study of algebraically closed fields emerged in the as mathematicians sought to understand the solvability of polynomial equations over different number systems, driven by the . This theorem asserts that every non-constant polynomial with coefficients has at least one root, thereby establishing the complex numbers as a field where all such polynomials factor completely into linear terms. demonstrated the in his 1799 doctoral dissertation at the University of Helmstedt, providing a geometric proof using the that, while groundbreaking, contained gaps by modern standards. He later offered additional proofs in 1816 (two versions) and 1849 to address these issues. The motivation intensified with Évariste Galois's groundbreaking work in the , which linked the solvability of polynomials by to the structure of field extensions, highlighting the need for fields where polynomials could be fully resolved without adjoining extraneous roots. Galois's theory demonstrated that certain quintic equations resist solutions due to the of their splitting fields, prompting deeper inquiry into extensions that "close" under root extraction. This framework underscored the desire for a universal field property ensuring all polynomials split completely, influencing subsequent efforts to generalize beyond the complexes. In the early , the transition to formalized these ideas, with Ernst Steinitz's seminal 1910 paper "Algebraische Theorie der Körper" providing the first axiomatic treatment of and proving that every admits an —an extension that is algebraically closed and unique up to over the base . Steinitz's relied on to adjoin roots iteratively, marking a pivotal shift from concrete number to abstract structures. Concurrently, advanced the formalization through her work on and theory, such as her 1921 paper on in commutative rings, which provided tools to analyze extensions within broader algebraic contexts. The term "algebraically closed field" gained prominence in the 1920s and 1930s amid developments in and , as seen in Bartel van der Waerden's 1930 textbook Moderne Algebra, which synthesized these concepts for a of mathematicians. This period solidified the notion's role in studying polynomial factorization universally, bridging classical equation-solving with modern abstract methods.

Examples

Complex Numbers

The complex numbers \mathbb{C} are constructed as the \mathbb{R}/(x^2 + 1), or equivalently, as the field extension \mathbb{R} where i is a satisfying i^2 = -1./06%3A_Complex_Numbers/6.01%3A_Complex_Numbers) This adjunction resolves the equation x^2 + 1 = 0 over the reals \mathbb{R}, which has no real solutions, and equips \mathbb{C} with the structure of a containing \mathbb{R} as a subfield. Elements of \mathbb{C} are of the form a + bi with a, b \in \mathbb{R}, and arithmetic operations are defined componentwise for the real and imaginary parts, inheriting field properties from \mathbb{R}. The field \mathbb{C} exemplifies an algebraically closed field, as established by the . This theorem asserts that every non-constant p(x) \in \mathbb{C} of degree n \geq 1 has exactly n roots in \mathbb{C}, counting multiplicities. In particular, if p(x) = \sum_{k=0}^n a_k x^k with a_n \neq 0 and \deg(p) = n, then p(x) = a_n \prod_{k=1}^n (x - r_k) for some r_1, \dots, r_n \in \mathbb{C}. A standard analytic proof begins by assuming, for contradiction, that a non-constant p(x) \in \mathbb{C} has no roots in \mathbb{C}. Then $1/p(z) is an (holomorphic on all of \mathbb{C}), as p has no zeros. For sufficiently large |z| = R, the leading term dominates, so |p(z)| \approx |a_n| R^n, implying |1/p(z)| < 1 outside a compact disk. Since $1/p is continuous and thus bounded inside the disk, it is bounded everywhere. By Liouville's theorem, a bounded is constant, so p must be constant, a contradiction. Finding one root allows factoring p(x) = (x - r) q(x) with \deg(q) = n-1, and induction yields the full factorization./05%3A_Cauchy_Integral_Formula/5.05%3A_Amazing_consequence_of_Cauchys_integral_formula) Carl Friedrich Gauss provided one of the first rigorous proofs of the fundamental theorem in his 1799 doctoral dissertation, using a topological argument based on the intermediate value theorem and curve intersections; he later refined it with additional proofs in 1816. Subsequent algebraic proofs were developed by Gauss and , leveraging field extensions and polynomial irreducibility. The field \mathbb{C} is precisely the algebraic closure of \mathbb{R}, meaning every algebraic extension of \mathbb{R} within \mathbb{C} is contained in \mathbb{C}, and \mathbb{C} is algebraic over \mathbb{R}. Up to isomorphism, \mathbb{C} is the unique algebraically closed field of characteristic $0and cardinality2^{\aleph_0}$ (the continuum), a consequence of the categoricity of the theory of algebraically closed fields in uncountable cardinalities.

Algebraic Closures

The algebraic closure of a field K, denoted \overline{K}, is defined as the smallest algebraically closed field that contains K as a subfield; equivalently, it is the algebraic field extension of K in which every polynomial in K splits into linear factors. This extension is algebraic over K, meaning that every element \alpha \in \overline{K} satisfies a polynomial equation with coefficients in K. In particular, for any such \alpha, the degree of the extension [K(\alpha):K] is finite, and the minimal polynomial of \alpha over K splits completely into linear factors within \overline{K}. Key properties of \overline{K} include its infinitude unless K is already algebraically closed, as adjoining roots iteratively extends beyond any finite degree if necessary. Moreover, any two algebraic closures of K are isomorphic as field extensions over K, ensuring a canonical structure up to this equivalence. For any field K, an algebraic closure exists and can be constructed as the direct limit of the directed system of all finite algebraic extensions of K, or alternatively by first adjoining a transcendence basis to form a purely transcendental extension and then taking its algebraic closure. A concrete example is the algebraic closure of the rational numbers \mathbb{Q}, denoted \overline{\mathbb{Q}}, which comprises all algebraic numbers—roots of non-constant polynomials with rational coefficients—and forms a countable field of characteristic zero. This field serves as a universal algebraic extension of \mathbb{Q}, embedding into the complex numbers while remaining strictly smaller.

Absence in Finite Fields

Finite fields, which are fields with a finite number of elements, cannot be algebraically closed. Suppose F_q is a finite field with q = p^n elements, where p is prime and n \geq 1. If F_q were algebraically closed, every non-constant polynomial in F_q would factor completely into linear factors over F_q, implying that F_q has no proper algebraic extensions. However, it is a well-established result that every finite field admits proper algebraic extensions of every positive integer degree m; specifically, there exists a unique (up to isomorphism) extension F_{q^m} of degree m over F_q, which properly contains F_q for m > 1. This contradicts the assumption, as the algebraic closure of F_q would be an infinite union of these finite extensions, properly extending F_q. The existence of such proper extensions stems from the presence of irreducible polynomials of every degree greater than 1 in F_q. For instance, the polynomial x^q - x factors completely over F_q as x^q - x = \prod_{a \in F_q} (x - a), accounting for all elements of F_q as roots. Yet, the ring F_q contains monic irreducible polynomials of each degree d \geq 2, whose roots generate the extensions F_{q^d}. The number of such irreducibles of degree d is given by \frac{1}{d} \sum_{e \mid d} \mu(e) q^{d/e} > 0, ensuring their existence. In the specific case of characteristic p, no finite field satisfies the condition that polynomials like the Artin-Schreier form x^p - x - a factor completely into linears without roots outside the field for all a. For appropriate choices of a \in F_q (where q is not a power of p in certain cases, but generally in char p), x^p - x - a is irreducible over F_q, generating a proper extension of degree p. This irreducibility follows from the fact that if it had a root \alpha \in F_q, then all translates \alpha + i for i = 1, \dots, p-1 would also be roots, implying complete splitting, but for suitable a, no roots exist in F_q. As a consequence, all algebraically closed fields of p must be . The \overline{\mathbb{F}_p} of the prime \mathbb{F}_p is a countable of the finite fields \mathbb{F}_{p^k} for k \geq 1.

Equivalent Characterizations

Polynomial Factorization Properties

A F is algebraically closed if and only if every nonconstant in F has a in F. This condition implies that the of any over F is F itself, as all roots lie within F. An equivalent characterization is that the only irreducible polynomials in F are the linear ones. If there existed an irreducible polynomial of degree greater than 1, adjoining a would yield a proper of F, contradicting algebraic closedness. Conversely, if all irreducibles are linear, every nonconstant factors into linears, ensuring all roots are in F. Another equivalent condition is that every polynomial of prime degree in F has a root in F. A of prime is either irreducible or has a linear ; irreducibility would imply a proper extension, which cannot occur in an algebraically closed field. This prevents the existence of irreducibles of prime degree and extends to all degrees by composition. In an algebraically closed field F, the F is a where every nonconstant factors uniquely (up to units and ordering) into linear factors, accounting for multiplicities. Specifically, for any p(x) \in F of degree n \geq 1, p(x) = c \prod_{i=1}^k (x - \alpha_i)^{m_i}, where c \in F is the leading coefficient, the \alpha_i \in F are distinct , the multiplicities m_i \geq 1 are positive integers, and \sum m_i = n. This factorization follows by on the degree n. For n=1, p(x) is already linear. Assume true for degrees less than n; since F is algebraically closed, p(x) has a \alpha_1 \in F, so p(x) = (x - \alpha_1) q(x) with \deg q = n-1. By the inductive , q(x) factors into linears, and the full of p(x) follows, with multiplicities determined via the for repeated . Uniqueness holds because F is a UFD and the linear factors are irreducible.

Field Extension Properties

A field F is algebraically closed if and only if it has no proper algebraic extensions, meaning that every algebraic extension E of F satisfies E = F. This characterization implies that every element algebraic over F already belongs to F, as adjoining any such element outside F would yield a proper algebraic extension. Equivalently, an algebraically closed field F admits no proper finite field extensions, since all finite extensions are algebraic./21%3A_Fields/21.01%3A_Extension_Fields) As a consequence, any proper extension of F must be transcendental, involving elements not algebraic over F, and F is maximal among algebraic fields in the sense that no larger field containing F can be algebraic over it. This extension property is closely tied to the over F: F is algebraically closed if and only if every in F splits completely into linear factors in F. To see the connection to extensions, suppose toward contradiction that L/F is a proper . Let \alpha \in L \setminus F; then the minimal polynomial m(x) of \alpha over F is of degree at least 2. However, since F is algebraically closed, m(x) must split into linear factors in F, contradicting the irreducibility of m(x) unless its degree is 1 (which would imply \alpha \in F). Thus, no such proper exists, and the simple extensions F(\alpha)/F for algebraic \alpha are trivial. In fields of positive characteristic p, algebraically closed fields have the additional property that all purely inseparable extensions are trivial. This follows because such fields are perfect, meaning every element of F is a p-th power in F, precluding nontrivial purely inseparable extensions of degree p^k > 1 for k \geq 1.

Linear Algebra Properties

A F is algebraically closed every F-linear of the finite-dimensional F^n (for any n \geq 1) admits an eigenvalue in F, meaning every in M_n(F) has a characteristic root in F. To see that algebraic closedness implies the existence of eigenvalues, consider a linear T: F^n \to F^n. The \chi_T(\lambda) = \det(\lambda I - T) is a of degree n \geq 1 with coefficients in F. Since F is algebraically closed, \chi_T(\lambda) has a \lambda_1 \in F, which serves as an eigenvalue of T. More generally, because F is algebraically closed, the splits completely over F: \chi_T(\lambda) = \prod_{i=1}^n (\lambda - \lambda_i), where each \lambda_i \in F is an eigenvalue of T (counted with algebraic multiplicity). The eigenvalues are precisely the roots of \chi_T(\lambda), establishing the connection between polynomial roots in F and the spectrum of endomorphisms. For the converse, suppose every matrix in M_n(F) has an eigenvalue in F for all n \geq 1. Consider an arbitrary non-constant monic polynomial p(\lambda) \in F[\lambda] of degree d \geq 1. The companion matrix C(p) \in M_d(F) associated to p(\lambda) has characteristic polynomial exactly p(\lambda). Thus, p(\lambda) has a root in F. Any non-constant polynomial in F[\lambda] is associate to a monic one, so every such polynomial has a root in F, implying F is algebraically closed. This characterization has significant implications for the structure of vector spaces over F. Since the splits completely into linear factors over F, the space F^n decomposes as a of generalized eigenspaces corresponding to the eigenvalues \lambda_i: F^n = \bigoplus_{i=1}^k \ker \left( (T - \lambda_i I)^{m_i} \right), where m_i is the algebraic multiplicity of \lambda_i, and each generalized eigenspace is under T. This reflects the field's completeness, enabling the Jordan canonical form for any . The property extends to the representation of s as s over the F. Finite-dimensional vector spaces over F with endomorphism T correspond to finitely generated torsion F-s via the action of x as multiplication by T. Over an algebraically closed field, the primary decomposition theorem yields that every such decomposes as a of cyclic s, each annihilated by a power of a linear [factor (x](/page/Factor_X) - \lambda) with \lambda \in F.

Constructions

Existence of Algebraic Closures

The existence of an algebraic closure for every field is a fundamental result in field theory. For any field K, an algebraic closure \overline{K} is an algebraic field extension of K that is itself algebraically closed, satisfying K \subseteq \overline{K} and the property that every \alpha \in \overline{K} is algebraic over K, meaning \alpha satisfies a non-constant polynomial equation with coefficients in K. This existence was established by Ernst Steinitz in 1910, who proved that every K admits an \overline{K}, unique up to over K. Steinitz's theorem provides the cornerstone for studying algebraic extensions, ensuring that such closures exist without relying on specific constructions for particular fields. One standard proof of existence uses from . Consider the set \mathcal{E} of all algebraic field extensions of K, partially ordered by inclusion. This poset is nonempty (as K itself is in \mathcal{E}) and inductive, since the union of any chain of algebraic extensions is again an algebraic extension of K. By , \mathcal{E} has a maximal element L, which must be algebraically closed: if not, adjoining a root of some over L would yield a strictly larger algebraic extension, contradicting maximality. An alternative, more constructive proof avoids full reliance on Zorn's lemma by building the closure via transfinite induction or direct limits. Enumerate all monic irreducible polynomials over K as \{f_\alpha \mid \alpha < \kappa\}, where \kappa is the cardinality of K. Start with K_0 = K, and at each successor stage \beta + 1, adjoin all roots of f_\beta to K_\beta to form K_{\beta+1}; at limit ordinals, take the direct limit. The resulting field at stage \kappa is an algebraic closure of K. This approach uses the axiom of choice only for well-ordering but provides an explicit iterative construction. The proofs must account for the characteristic of the field. In characteristic 0, all algebraic extensions are separable, so the above constructions directly yield the . In characteristic p > 0, inseparable extensions may arise, so first form the perfect closure K^{1/p^\infty} by iteratively adjoining p-th roots of all elements until every element has a p-th root; this has the same as K, reducing the problem to the perfect case where extensions are separable. Specific examples illustrate cardinality aspects: the algebraic closure \overline{\mathbb{Q}} of the rationals \mathbb{Q} is countable, as it is the union of the splitting fields of all countably many polynomials over \mathbb{Q}, each of finite degree. In contrast, the algebraic closure \overline{\mathbb{R}} of the reals \mathbb{R} is the complex numbers \mathbb{C}, which is uncountable.

Uniqueness up to Isomorphism

An algebraically closed field L is called an algebraic closure of a field K if L is algebraic over K. If L and M are algebraic closures of the same field K, then there exists a K-isomorphism L \cong M. This uniqueness up to isomorphism over K was established by Steinitz in 1910. The proof proceeds by showing that any two such closures are isomorphic over K. Since both L and M are algebraically closed and algebraic over K, there exists a K-embedding i: L \hookrightarrow M. As M is algebraically closed, and L algebraic over K implies no proper algebraic extensions within M, the embedding i must be surjective, hence an isomorphism. The non-uniqueness of the isomorphism arises from the possible K-automorphisms of M. A key implication of this uniqueness is that all algebraic closures of K share the same absolute Galois group up to isomorphism, defined as \mathrm{Gal}(K^\mathrm{sep}/K), where K^\mathrm{sep} is the separable closure of K contained in any algebraic closure. This group captures the non-uniqueness of the isomorphisms between closures. While the full uniqueness theorem relies on (or the ), separable closures in characteristic zero are unique up to isomorphism without choice, as separability coincides with algebraicity there; in general, however, choice is necessary. In characteristic p > 0, ensuring the closure is perfect (i.e., every element has a p-th root) guarantees the isomorphism property holds uniformly. For a concrete example, all algebraic closures of the function field \mathbb{F}_p(t) over \mathbb{F}_p are isomorphic over \mathbb{F}_p(t), reflecting the general despite the transcendental nature of the base .

Additional Properties

Rational Function Decompositions

In an algebraically closed F, the field of F(x) admits a unique for any expressed as f = p/q, where p and q are coprime in F with q \neq 0. This decomposition takes the form f(x) = h(x) + \sum_{\alpha \in F} \sum_{k=1}^{m_\alpha} \frac{c_{\alpha,k}}{(x - \alpha)^k}, where h(x) is a in F (the polynomial part obtained via if \deg p \geq \deg q), the \alpha are the distinct of q(x) in F, m_\alpha is the multiplicity of \alpha, and the coefficients c_{\alpha,k} \in F are uniquely determined. The existence and uniqueness of this decomposition follow from the fact that F is algebraically closed, which ensures that q(x) factors completely into linear factors over F, combined with the unique factorization domain structure of F. Specifically, once q(x) = c \prod_{\alpha} (x - \alpha)^{m_\alpha} with c \in F^\times, the partial fractions arise by expressing f(x) as a sum over these irreducible linear factors raised to their multiplicities, leveraging to ensure coprimality allows separation of terms. A proof proceeds by first performing polynomial division to isolate h(x), then factoring q(x) completely; the remaining proper fraction is decomposed using the Heaviside method (or coefficient comparison) to solve for residues, with uniqueness guaranteed by the of the basis \{1/(x - \alpha)^k\} over the poles. For a simple pole at \alpha (i.e., m_\alpha = 1), the residue simplifies to c_\alpha = \frac{p(\alpha)}{q'(\alpha)}, derived by evaluating the as x \to \alpha or directly from the cover-up formula. This algebraic generalizes the classical version over the real or complex numbers to any algebraically closed field, providing a purely algebraic tool without reliance on or . As a consequence, every of F(x) admits a Laurent series-like at each finite \alpha \in F, facilitating computations in field extensions and over F.

Coprime Polynomial Behaviors

In an algebraically closed field F, two polynomials f, g \in F are coprime, meaning their is 1, they share no common in F. This follows from the fact that every in F is linear, so any common root would imply a shared linear , contradicting coprimeness. The generated by coprime polynomials f and g, denoted (f, g), equals the unit ideal (1) in F, which implies that the set of common zeros of f and g is empty. This is a direct consequence of the weak form of , which states that over an algebraically closed field, a proper corresponds to a nonempty , so the unit ideal corresponds to the empty variety. A key characterization is that in an algebraically closed field F, two polynomials are coprime if and only if their is nonzero, which occurs precisely when their sets of are disjoint. The \operatorname{Res}(f, g) can be expressed as \operatorname{Res}(f, g) = a^m \prod_{i=1}^n g(\alpha_i), where a is the leading coefficient of f, m = \deg g, n = \deg f, and \alpha_i are the of f; equivalently, up to sign and leading coefficients, it is \prod_j f(\beta_j) over the \beta_j of g, and this product is nonzero if and only if no \beta_j is a of f. For coprime f, g \in F, holds: there exist polynomials a, b \in F such that a f + b g = 1. This is a consequence of F being a , where coprime elements generate the unit ideal. Over the complex numbers \mathbb{C}, an algebraically closed field, coprime polynomials define disjoint zero sets on the affine line \mathbb{A}^1(\mathbb{C}), ensuring no overlap in their root loci.