12: Field theory and polynomials
Field theory considers sets, such as the real number line, on which all the usual arithmetic properties hold --- those governing addition, subtraction, multiplication and division. The study of multiple fields through Galois theory is important for the study of polynomial equations, and thus has applications to number theory and group theory. Tools in this area can also be used to show certain Euclidean geometry constructions are impossible, and that certain functions have no elementary antiderivative!
Specifically, a field is a commutative ring in which every nonzero element is assumed to have a multiplicative inverse. Examples include the real number field R, the complex numbers C, the rational numbers Q, finite fields (the Galois fields with p^n elements for some prime p), the p-adic numbers Q_p, and various fields of functions such as the collection of meromorphic entire functions. All these examples are of characteristic zero except the finite fields (if there is a finite set of 1's which add to zero, the cardinality of the smallest such set is the characteristic, a prime).
Several constructions allow the creation of more fields, and in particular generate field extensions K/F (i.e. nested pairs of fields F < K). Algebraic extensions are those in which every element of K is the root of a polynomial with coefficients in F; elements of K which are not algebraic are transcendental over F (e.g. pi is transcendental over Q). Thus C is an algebraic extension of R, and the field of rational functions F(x) in one variable is transcendental over F. If P(x) is an irreducible polynomial in the ring F[x] then the quotient ring K = F[x]/( P(x) ) is an algebraic extension of F in which P has a root.
Some themes of field theory are then immediately apparent.
First, the study of fields is the appropriate venue for the consideration of some topics in number theory. For example, one approach to Fermat's Last Theorem (that the equation x^n + y^n = z^n has no solutions in positive integers when n is greater than 2) suggests factoring numbers arising from a putative solution: x^n+y^n=(x+y)(x+ry)...(x+sy) where r,...,s are roots of unity lying in an extension field of Q. This analysis centers upon the ring of integers in the extension field (which is a well-defined subring of number fields) moreso than the extension field itself; thus this discussion is more appropriately considered part of Number Theory than Field Theory, but certain one uses tools from Field Theory -- the norm and trace mappings, the structure of the group of units, and so on. The Kronecker-Weber theorem (classifying abelian extensions of Q) and Hilbert's Theorem 90 (norms in cyclic extensions) are arguably Field Theory but perhaps more at home in Algebraic Number Theory.
Likewise one might include in Field Theory topics in local fields or finite fields, but the current Mathematics Subject Classification places these with Number Theory. Here a local field is the quotient field of a ring with a unique maximal ideal (such as a power series ring); thus for example the field of Laurent series R((x)) is a local field, as is a p-adic field Q_p.
Setting aside the themes in number theory, the most significant topic in Field Theory proper involves the study of field extensions K/F, in particular Galois theory. Interesting examples of field extensions involve splitting fields of polynomials (the smallest extension in which a given polynomial factors into linear factors), the algebraic closure (the compositum of all such splitting fields), purely transcendental extensions (in the Weil formulation of Algebraic Geometry, this is the setting for "generic points" of algebraic varieties), and (in)separable extensions (relevant only for infinite fields of non-zero characteristic.) An interesting example of field extensions is the set of "constructible" real numbers; this gives tools sufficient to prove that one cannot duplicate the cube, and in general allows an analysis of other construction problems in Euclidean Geometry.
Given a field extension K/F, we consider the automorphisms of the field K which act as the identity map on F; for example complex conjugation is such an automorphism for the field extension C/R. These automorphisms form a group, the Galois group Gal(K/F), and for sufficiently well-behaved field extensions this group "controls" the group extension (e.g. it enumerates the intermediate extension fields by pairing them off with subgroups of Gal(K/F).) The Galois group of a polynomial is the group of its splitting field; this is the tool used by Galois to establish that the quintic cannot be solved in radicals. The absolute Galois group of F is the Galois group of its algebraic closure; this is generally a complicated structure (e.g. it is more appropriately treated as a topological group) but in principle many number-theoretic problems can be reduced to its study (e.g. Galois cohomology is applied in arithmetic algebraic geometry).
The field of meromorphic functions on a given complex domain comes equipped with an additional operator, differentiation. Differential algebra is the study of such fields, suitably axiomatized. This provides a setting in which it is possible to study whether or not a differential equation with coefficients lying in one differential field have solutions within that field (Picard-Vessiot theory). In particular, one can decide (Liouville's theorem, Risch's algorithm) whether the equation dy/dt=f(t) has an "elementary" solution when f(t) is an "elementary" function (e.g. rational, or a rational combination of exponential functions). In addition, it is possible to consider differential equations over other fields than the real and complex numbers, leading to p-adic differential equations.
A different additional structure which may be placed on a field is a topology, and in particular a metric, a valuation, or an ordering. In addition to the usual metric inherited from the real line, the rational numbers have p-adic metrics for each prime p; generalizations exist for number fields, as well as for fields of functions and formal series. The appropriate field extensions in this area are those in which the topology or metric can also be extended, and in particular one may study the topological completion of the field (e.g. the real line is a completion of Q). The topological fields which are locally compact are known (essentially the completions of number fields). A number of topics in algebra can be studied more simply upon passing to a complete field, and in some cases there is a "local-to-global principle" which guarantees the problem can be completely analyzed in this way.
The fields R and C are remarkable in several ways: C is algebraically closed (the Fundamental Theorem of Algebra), R is topologically complete, and is linearly ordered. Naturally one investigates generalizations (e.g. the Artin-Schreier theorem on subfields of finite index in algebraically closed fields). In addition to the obvious applications to analysis, these fields are particularly valuable because they admit a characterization by first-order axioms; Tarski's Elimination of Quantifiers allows us in principle to resolve field-theoretic questions concerning these fields. By applying results in Model Theory to these axioms one may construct non-standard models of the real field, leading to Non-Standard Analysis (in which, in particular, arguments of calculus can be carried out with reference to epsilons but not limits!).
If the existence of multiplicative inverses is not assumed, the remaining field axioms describe commutative rings, discussed on a separate page (13-XX). Note that extension fields K/F are actually commutative algebras over F, as are the polynomial rings F[x] mentioned repeatedly above. Topics in commutative algebras over a field F are frequently addressed by studying the corresponding algebras over extension fields K/F.
If instead inverses are assumed but the commutative law is not, the field axioms describe division algebras; there is a separate page with a long FAQ about these and related topics; as appropriate, this material is a topic in 16: Noncommutative Ring Theory or 17: Nonassociative Ring Theory. (Note that some writers, particularly in French, use the term "field" (corps) to include non-commutative division algebras by default; these authors specify commutativity when referring to the objects under study in this page.)
For fields of interest in 10,11: Number Theory, see sections 11R (number fields), 11S (local fields), and 11T (finite fields). For Galois theory applied specifically to algebraic number fields, see especially sections 11R32 and 11S20.
Since Galois theory establishes a connection between field extensions and symmetry groups, topics in this section often lead to connections with 20: Group Theory, e.g. the Inverse Galois Problem (whether or not a given group is the Galois group of some number field.) There are also groups associated with fields in other ways, e.g. the additive group of F, the multiplicative group F*, the general linear groups GL_n(F), and other algebraic groups, but these topics are not typically associated to section 12. (For algebraic groups in general see sections 20 or 22; for matrix groups see 15: Linear algebra; for matrix groups over finite fields see 20D: Finite group theory.)
Fields of functions of algebraic varieties (essentially the quotient fields of rings F[x1,...,xn]/(P) where P is a multivariable polynomial) are more properly treated in 14: Algebraic Geometry, although these are really just discussions of fields of finite transcendence degree over the ground field. (Also note that arithmetic questions in algebraic geometry -- e.g. are there rational points on a certain variety -- are productively addressed by analyzing the action of the absolute Galois group on the set of points on the variety as defined over the algebraic closure of the original field.)
Likewise fields of meromorphic functions and local rings of germs of functions are usually treated with their applications to 30: Complex Analysis, 32: Several complex variables, and 58: Analysis on manifolds.
For topics in 51: Euclidean geometry, see especially 51M15: Euclidean constructions. Note that any field can be used for coordinates of geometries; properties of the field are often reflected in properties of the geometry (e.g. Pappus's theorem, Desargues's theorem). This material is usually classified with geometry.
The study of real closed fields was first advanced by those working in 03: Mathematical Logic; this topic has extended to questions of computability and even to the development of algorithms; see 68: Computer Science and 13P: Computational problems in commutative algebra.
For tools in homological algebra which can be applied here, see 18:Category theory and homological algebra.
As noted, Field Theory has applications to several famous problems and involves a well-known romantic figure; see 01: History and Biography for more information.
"Field theory" in this sense has nothing to do with the potential fields in mathematical physics.
This is among the smaller areas in the Math Reviews database.
Until section Number Theory was reorganized (in 1984) from section 10 to section 11, some of the topics now considered number theory were classified with section 12. These are shown in the accompanying diagram as
Browse all (old) classifications for this area at the AMS.
Most undergraduate and graduate textbooks in abstract algebra include a portion on fields, usually at least including some Galois Theory and perhaps proofs of some of the famous results listed above. Others focus narrowly on Galois Theory, say, and so are more appropriately mentioned on the index page for the appropriate subdiscipline. We mention here only those texts which cover Field Theory more or less as understood here. Classic, accessible texts:
Collins, G. E.: "Quantifier elimination for real closed fields: a guide to the literature" Computer algebra, 79--81, Springer, Vienna, 1983.
There is quite a bit of software for algebraic number fields, including computations with units, class groups, Galois actions, and so on. See the pages for 11: Number Theory.