Classic definition

Formally, a field is a set F together with two binary operations on F called addition and multiplication.^[1] A binary operation on F is a mapping F × F → F, that is, a correspondence that associates with each ordered pair of elements of F a uniquely determined element of F.^[2][3] The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted ab or a ⋅ b. These operations are required to satisfy the following properties, referred to as field axioms (in these axioms, a, b, and c are arbitrary elements of the field F):

• Associativity of addition and multiplication: a + (b + c) = (a + b) + c, and a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c.
• Commutativity of addition and multiplication: a + b = b + a, and a ⋅ b = b ⋅ a.
• Additive and multiplicative identity: there exist two distinct elements 0 and 1 in F such that a + 0 = a and a ⋅ 1 = a.
• Additive inverses: for every a in F, there exists an element in F, denoted −a, called the additive inverse of a, such that a + (−a) = 0.
• Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by a⁻¹ or 1/a, called the multiplicative inverse of a, such that a ⋅ a⁻¹ = 1.
• Distributivity of multiplication over addition: a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c).

An equivalent, and more succinct, definition is: a field has two commutative operations, called addition and multiplication; it is a group under addition with 0 as the additive identity; the nonzero elements are a group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition.

Even more succinctly: a field is a commutative ring where 0 ≠ 1 and all nonzero elements are invertible under multiplication.

Alternative definition

Fields can also be defined in different, but equivalent ways. One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties. Division by zero is, by definition, excluded.^[4] In order to avoid existential quantifiers, fields can be defined by two binary operations (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and two nullary operations (the constants 0 and 1). These operations are then subject to the conditions above. Avoiding existential quantifiers is important in constructive mathematics and computing.^[5] One may equivalently define a field by the same two binary operations, one unary operation (the multiplicative inverse), and two (not necessarily distinct) constants 1 and −1, since 0 = 1 + (−1) and −a = (−1)a.^{[lower-alpha 1]}

Rational numbers

Rational numbers have been widely used a long time before the elaboration of the concept of field. They are numbers that can be written as fractions a/b, where a and b are integers, and b ≠ 0. The additive inverse of such a fraction is −a/b, and the multiplicative inverse (provided that a ≠ 0) is b/a, which can be seen as follows:

${\displaystyle {\frac {b}{a}}\cdot {\frac {a}{b}}={\frac {ba}{ab}}=1.}$

The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows:^[6]

${\displaystyle {\begin{aligned}&{\frac {a}{b}}\cdot \left({\frac {c}{d}}+{\frac {e}{f}}\right)\\[6pt]={}&{\frac {a}{b}}\cdot \left({\frac {c}{d}}\cdot {\frac {f}{f}}+{\frac {e}{f}}\cdot {\frac {d}{d}}\right)\\[6pt]={}&{\frac {a}{b}}\cdot \left({\frac {cf}{df}}+{\frac {ed}{fd}}\right)={\frac {a}{b}}\cdot {\frac {cf+ed}{df}}\\[6pt]={}&{\frac {a(cf+ed)}{bdf}}={\frac {acf}{bdf}}+{\frac {aed}{bdf}}={\frac {ac}{bd}}+{\frac {ae}{bf}}\\[6pt]={}&{\frac {a}{b}}\cdot {\frac {c}{d}}+{\frac {a}{b}}\cdot {\frac {e}{f}}.\end{aligned}}}$

Real and complex numbers

The multiplication of complex numbers can be visualized geometrically by rotations and scalings.

The real numbers R, with the usual operations of addition and multiplication, also form a field. The complex numbers C consist of expressions

a + bi, with a, b real,

where i is the imaginary unit, i.e., a (non-real) number satisfying i² = −1. Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold for C. For example, the distributive law enforces

(a + bi)(c + di) = ac + bci + adi + bdi² = (ac − bd) + (bc + ad)i.

It is immediate that this is again an expression of the above type, and so the complex numbers form a field. Complex numbers can be geometrically represented as points in the plane, with Cartesian coordinates given by the real numbers of their describing expression, or as the arrows from the origin to these points, specified by their length and an angle enclosed with some distinct direction. Addition then corresponds to combining the arrows to the intuitive parallelogram (adding the Cartesian coordinates), and the multiplication is – less intuitively – combining rotating and scaling of the arrows (adding the angles and multiplying the lengths). The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.

Constructible numbers

The geometric mean theorem asserts that h² = pq. Choosing q = 1 allows construction of the square root of a given constructible number p.

In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with compass and straightedge. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field of constructible numbers.^[7] Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only compass and straightedge. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field Q of rational numbers. The illustration shows the construction of square roots of constructible numbers, not necessarily contained within Q. Using the labeling in the illustration, construct the segments AB, BD, and a semicircle over AD (center at the midpoint C), which intersects the perpendicular line through B in a point F, at a distance of exactly ${\displaystyle h={\sqrt {p}}}$ from B when BD has length one.

Not all real numbers are constructible. It can be shown that ${\displaystyle {\sqrt[{3}]{2}}}$ is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a cube with volume 2, another problem posed by the ancient Greeks.

A field with four elements

In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called O, I, A, and B. The notation is chosen such that O plays the role of the additive identity element (denoted 0 in the axioms above), and I is the multiplicative identity (denoted 1 in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,

A ⋅ (B + A) = A ⋅ I = A, which equals A ⋅ B + A ⋅ A = I + B = A, as required by the distributivity.

This field is called a finite field or Galois field with four elements, and is denoted F₄ or GF(4).^[8] The subset consisting of O and I (highlighted in red in the tables at the right) is also a field, known as the binary field F₂ or GF(2). In the context of computer science and Boolean algebra, O and I are often denoted respectively by false and true, and the addition is then denoted XOR (exclusive or). In other words, the structure of the binary field is the basic structure that allows computing with bits.

Consequences of the definition

One has a ⋅ 0 = 0 and −a = (−1) ⋅ a. In particular, one may deduce the additive inverse of every element as soon as one knows −1.^[9]

If ab = 0 then a or b must be 0, since, if a ≠ 0, then b = (a⁻¹a)b = a⁻¹(ab) = a⁻¹ ⋅ 0 = 0. This means that every field is an integral domain.

In addition, the following properties are true for any elements a and b:

−0 = 0

1⁻¹ = 1

(−(−a)) = a

(−a) ⋅ b = a ⋅ (−b) = −(a ⋅ b)

(a⁻¹)⁻¹ = a if a ≠ 0

Additive and multiplicative groups of a field

The axioms of a field F imply that it is an abelian group under addition. This group is called the additive group of the field, and is sometimes denoted by (F, +) when denoting it simply as F could be confusing.

Similarly, the nonzero elements of F form an abelian group under multiplication, called the multiplicative group, and denoted by (F ∖ {0}, ⋅) or just F ∖ {0}, or F^×.

A field may thus be defined as set F equipped with two operations denoted as an addition and a multiplication such that F is an abelian group under addition, F ∖ {0} is an abelian group under multiplication (where 0 is the identity element of the addition), and multiplication is distributive over addition.^{[lower-alpha 2]} Some elementary statements about fields can therefore be obtained by applying general facts of groups. For example, the additive and multiplicative inverses −a and a⁻¹ are uniquely determined by a.

The requirement 1 ≠ 0 is imposed by convention to exclude the trivial ring, which consists of a single element; this guides any choice of the axioms that define fields.

Every finite subgroup of the multiplicative group of a field is cyclic (see Root of unity § Cyclic groups).

Characteristic

In addition to the multiplication of two elements of F, it is possible to define the product n ⋅ a of an arbitrary element a of F by a positive integer n to be the n-fold sum

a + a + ⋯ + a (which is an element of F.)

If there is no positive integer such that

n ⋅ 1 = 0,

then F is said to have characteristic 0.^[11] For example, the field of rational numbers Q has characteristic 0 since no positive integer n is zero. Otherwise, if there is a positive integer n satisfying this equation, the smallest such positive integer can be shown to be a prime number. It is usually denoted by p and the field is said to have characteristic p then. For example, the field F₄ has characteristic 2 since (in the notation of the above addition table) I + I = O.

If F has characteristic p, then p ⋅ a = 0 for all a in F. This implies that

(a + b)^p = a^p + b^p,

since all other binomial coefficients appearing in the binomial formula are divisible by p. Here, a^p := a ⋅ a ⋅ ⋯ ⋅ a (p factors) is the pth power, i.e., the p-fold product of the element a. Therefore, the Frobenius map

F → F : x ↦ x^p

is compatible with the addition in F (and also with the multiplication), and is therefore a field homomorphism.^[12] The existence of this homomorphism makes fields in characteristic p quite different from fields of characteristic 0.

Subfields and prime fields

A subfield E of a field F is a subset of F that is a field with respect to the field operations of F. Equivalently E is a subset of F that contains 1, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that 1 ∊ E, that for all a, b ∊ E both a + b and a ⋅ b are in E, and that for all a ≠ 0 in E, both −a and 1/a are in E.

Field homomorphisms are maps φ: E → F between two fields such that φ(e₁ + e₂) = φ(e₁) + φ(e₂), φ(e₁e₂) = φ(e₁) φ(e₂), and φ(1_E) = 1_F, where e₁ and e₂ are arbitrary elements of E. All field homomorphisms are injective.^[13] If φ is also surjective, it is called an isomorphism (or the fields E and F are called isomorphic).

A field is called a prime field if it has no proper (i.e., strictly smaller) subfields. Any field F contains a prime field. If the characteristic of F is p (a prime number), the prime field is isomorphic to the finite field F_p introduced below. Otherwise the prime field is isomorphic to Q.^[14]

Constructing fields from rings

A commutative ring is a set that is equipped with an addition and multiplication operation and satisfes all the axioms of a field, except for the existence of multiplicative inverses a⁻¹.^[26] For example, the integers Z form a commutative ring, but not a field: the reciprocal of an integer n is not itself an integer, unless n = ±1.

In the hierarchy of algebraic structures fields can be characterized as the commutative rings R in which every nonzero element is a unit (which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct ideals, (0) and R. Fields are also precisely the commutative rings in which (0) is the only prime ideal.

Given a commutative ring R, there are two ways to construct a field related to R, i.e., two ways of modifying R such that all nonzero elements become invertible: forming the field of fractions, and forming residue fields. The field of fractions of Z is Q, the rationals, while the residue fields of Z are the finite fields F_p.

Constructing fields within a bigger field

Fields can be constructed inside a given bigger container field. Suppose given a field E, and a field F containing E as a subfield. For any element x of F, there is a smallest subfield of F containing E and x, called the subfield of F generated by x and denoted E(x).^[29] The passage from E to E(x) is referred to by adjoining an element to E. More generally, for a subset S ⊂ F, there is a minimal subfield of F containing E and S, denoted by E(S).

The compositum of two subfields E and E′ of some field F is the smallest subfield of F containing both E and E′. The compositum can be used to construct the biggest subfield of F satisfying a certain property, for example the biggest subfield of F, which is, in the language introduced below, algebraic over E.^{[lower-alpha 3]}

Field extensions

The notion of a subfield E ⊂ F can also be regarded from the opposite point of view, by referring to F being a field extension (or just extension) of E, denoted by

F / E,

and read "F over E".

A basic datum of a field extension is its degree [F : E], i.e., the dimension of F as an E-vector space. It satisfies the formula^[30]

[G : E] = [G : F] [F : E].

Extensions whose degree is finite are referred to as finite extensions. The extensions C / R and F₄ / F₂ are of degree 2, whereas R / Q is an infinite extension.

Closure operations

A field is algebraically closed if it does not have any strictly bigger algebraic extensions or, equivalently, if any polynomial equation

f_n xⁿ + f_n−1xⁿ⁻¹ + ⋯ + f₁x + f₀ = 0, with coefficients f_n, ..., f₀ ∈ F, n > 0,

has a solution x ∊ F.^[33] By the fundamental theorem of algebra, C is algebraically closed, i.e., any polynomial equation with complex coefficients has a complex solution. The rational and the real numbers are not algebraically closed since the equation

x² + 1 = 0

does not have any rational or real solution. A field containing F is called an algebraic closure of F if it is algebraic over F (roughly speaking, not too big compared to F) and is algebraically closed (big enough to contain solutions of all polynomial equations).

By the above, C is an algebraic closure of R. The situation that the algebraic closure is a finite extension of the field F is quite special: by the Artin–Schreier theorem, the degree of this extension is necessarily 2, and F is elementarily equivalent to R. Such fields are also known as real closed fields.

Any field F has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as the algebraic closure and denoted F. For example, the algebraic closure Q of Q is called the field of algebraic numbers. The field F is usually rather implicit since its construction requires the ultrafilter lemma, a set-theoretic axiom that is weaker than the axiom of choice.^[34] In this regard, the algebraic closure of F_q, is exceptionally simple. It is the union of the finite fields containing F_q (the ones of order qⁿ). For any algebraically closed field F of characteristic 0, the algebraic closure of the field F((t)) of Laurent series is the field of Puiseux series, obtained by adjoining roots of t.^[35]

Ordered fields

A field F is called an ordered field if any two elements can be compared, so that x + y ≥ 0 and xy ≥ 0 whenever x ≥ 0 and y ≥ 0. For example, the real numbers form an ordered field, with the usual ordering ≥. The Artin–Schreier theorem states that a field can be ordered if and only if it is a formally real field, which means that any quadratic equation

${\displaystyle x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}=0}$

only has the solution x₁ = x₂ = ⋯ = x_n = 0.^[36] The set of all possible orders on a fixed field F is isomorphic to the set of ring homomorphisms from the Witt ring W(F) of quadratic forms over F, to Z.^[37]

An Archimedean field is an ordered field such that for each element there exists a finite expression

1 + 1 + ⋯ + 1

whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no infinitesimals (elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield of R.

Each bounded real set has a least upper bound.

An ordered field is Dedekind-complete if all upper bounds, lower bounds (see Dedekind cut) and limits, which should exist, do exist. More formally, each bounded subset of F is required to have a least upper bound. Any complete field is necessarily Archimedean,^[38] since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence 1/2, 1/3, 1/4, ..., every element of which is greater than every infinitesimal, has no limit.

Since every proper subfield of the reals also contains such gaps, R is the unique complete ordered field, up to isomorphism.^[39] Several foundational results in calculus follow directly from this characterization of the reals.

The hyperreals R^* form an ordered field that is not Archimedean. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis of non-standard analysis.

Topological fields

Another refinement of the notion of a field is a topological field, in which the set F is a topological space, such that all operations of the field (addition, multiplication, the maps a ↦ −a and a ↦ a⁻¹) are continuous maps with respect to the topology of the space.^[40] The topology of all the fields discussed below is induced from a metric, i.e., a function

d : F × F → R,

that measures a distance between any two elements of F.

The completion of F is another field in which, informally speaking, the "gaps" in the original field F are filled, if there are any. For example, any irrational number x, such as x = √2, is a "gap" in the rationals Q in the sense that it is a real number that can be approximated arbitrarily closely by rational numbers p/q, in the sense that distance of x and p/q given by the absolute value |x − p/q| is as small as desired. The following table lists some examples of this construction. The fourth column shows an example of a zero sequence, i.e., a sequence whose limit (for n → ∞) is zero.

The field Q_p is used in number theory and p-adic analysis. The algebraic closure Q_p carries a unique norm extending the one on Q_p, but is not complete. The completion of this algebraic closure, however, is algebraically closed. Because of its rough analogy to the complex numbers, it is sometimes called the field of complex p-adic numbers and is denoted by C_p.^[41]

Differential fields

Differential fields are fields equipped with a derivation, i.e., allow to take derivatives of elements in the field.^[44] For example, the field R(X), together with the standard derivative of polynomials forms a differential field. These fields are central to differential Galois theory, a variant of Galois theory dealing with linear differential equations.

Model theory of fields

In model theory, a branch of mathematical logic, two fields E and F are called elementarily equivalent if every mathematical statement that is true for E is also true for F and conversely. The mathematical statements in question are required to be first-order sentences (involving 0, 1, the addition and multiplication). A typical example, for n > 0, n an integer, is

φ(E) = "any polynomial of degree n in E has a zero in E"

The set of such formulas for all n expresses that E is algebraically closed. The Lefschetz principle states that C is elementarily equivalent to any algebraically closed field F of characteristic zero. Moreover, any fixed statement φ holds in C if and only if it holds in any algebraically closed field of sufficiently high characteristic.^[50]

If U is an ultrafilter on a set I, and F_i is a field for every i in I, the ultraproduct of the F_i with respect to U is a field.^[51] It is denoted by

ulim_i→∞ F_i,

since it behaves in several ways as a limit of the fields F_i: Łoś's theorem states that any first order statement that holds for all but finitely many F_i, also holds for the ultraproduct. Applied to the above sentence φ, this shows that there is an isomorphism^{[lower-alpha 5]}

${\displaystyle \operatorname {ulim} _{p\to \infty }{\overline {\mathbf {F} }}_{p}\cong \mathbf {C} .}$

The Ax–Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts (in both cases over all primes p)

ulim_p Q_p ≅ ulim_p F_p((t)).

In addition, model theory also studies the logical properties of various other types of fields, such as real closed fields or exponential fields (which are equipped with an exponential function exp : F → F^×).^[52]

Absolute Galois group

For fields that are not algebraically closed (or not separably closed), the absolute Galois group Gal(F) is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs all finite separable extensions of F. By elementary means, the group Gal(F_q) can be shown to be the Prüfer group, the profinite completion of Z. This statement subsumes the fact that the only algebraic extensions of Gal(F_q) are the fields Gal(F_qⁿ) for n > 0, and that the Galois groups of these finite extensions are given by

Gal(F_qⁿ / F_q) = Z/nZ.

A description in terms of generators and relations is also known for the Galois groups of p-adic number fields (finite extensions of Q_p).^[53]

Representations of Galois groups and of related groups such as the Weil group are fundamental in many branches of arithmetic, such as the Langlands program. The cohomological study of such representations is done using Galois cohomology.^[54] For example, the Brauer group, which is classically defined as the group of central simple F-algebras, can be reinterpreted as a Galois cohomology group, namely

Br(F) = H²(F, G_m).

K-theory

Milnor K-theory is defined as

${\displaystyle K_{n}^{M}(F)=F^{\times }\otimes \cdots \otimes F^{\times }/\left\langle x\otimes (1-x)\mid x\in F\setminus \{0,1\}\right\rangle .}$

The norm residue isomorphism theorem, proved around 2000 by Vladimir Voevodsky, relates this to Galois cohomology by means of an isomorphism

${\displaystyle K_{n}^{M}(F)/p=H^{n}(F,\mu _{l}^{\otimes n}).}$

Algebraic K-theory is related to the group of invertible matrices with coefficients the given field. For example, the process of taking the determinant of an invertible matrix leads to an isomorphism K₁(F) = F^×. Matsumoto's theorem shows that K₂(F) agrees with K₂^M(F). In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general.

Linear algebra and commutative algebra

If a ≠ 0, then the equation

ax = b

has a unique solution x in a field F, namely ${\displaystyle x=a^{-1}b.}$ This immediate consequence of the definition of a field is fundamental in linear algebra. For example, it is an essential ingredient of Gaussian elimination and of the proof that any vector space has a basis.^[55]

The theory of modules (the analogue of vector spaces over rings instead of fields) is much more complicated, because the above equation may have several or no solutions. In particular systems of linear equations over a ring are much more difficult to solve than in the case of fields, even in the specially simple case of the ring Z of the integers.

Finite fields: cryptography and coding theory

The sum of three points P, Q, and R on an elliptic curve E (red) is zero if there is a line (blue) passing through these points.

A widely applied cryptographic routine uses the fact that discrete exponentiation, i.e., computing

aⁿ = a ⋅ a ⋅ ⋯ ⋅ a (n factors, for an integer n ≥ 1)

in a (large) finite field F_q can be performed much more efficiently than the discrete logarithm, which is the inverse operation, i.e., determining the solution n to an equation

aⁿ = b.

In elliptic curve cryptography, the multiplication in a finite field is replaced by the operation of adding points on an elliptic curve, i.e., the solutions of an equation of the form

y² = x³ + ax + b.

Finite fields are also used in coding theory and combinatorics.

Geometry: field of functions

A compact Riemann surface of genus two (two handles). The genus can be read off the field of meromorphic functions on the surface.

Functions on a suitable topological space X into a field F can be added and multiplied pointwise, e.g., the product of two functions is defined by the product of their values within the domain:

(f ⋅ g)(x) = f(x) ⋅ g(x).

This makes these functions a F-commutative algebra.

For having a field of functions, one must consider algebras of functions that are integral domains. In this case the ratios of two functions, i.e., expressions of the form

${\displaystyle {\frac {f(x)}{g(x)}},}$

form a field, called field of functions.

This occurs in two main cases. When X is a complex manifold X. In this case, one considers the algebra of holomorphic functions, i.e., complex differentiable functions. Their ratios form the field of meromorphic functions on X.

The function field of an algebraic variety X (a geometric object defined as the common zeros of polynomial equations) consists of ratios of regular functions, i.e., ratios of polynomial functions on the variety. The function field of the n-dimensional space over a field F is F(x₁, ..., x_n), i.e., the field consisting of ratios of polynomials in n indeterminates. The function field of X is the same as the one of any open dense subvariety. In other words, the function field is insensitive to replacing X by a (slightly) smaller subvariety.

The function field is invariant under isomorphism and birational equivalence of varieties. It is therefore an important tool for the study of abstract algebraic varieties and for the classification of algebraic varieties. For example, the dimension, which equals the transcendence degree of F(X), is invariant under birational equivalence.^[56] For curves (i.e., the dimension is one), the function field F(X) is very close to X: if X is smooth and proper (the analogue of being compact), X can be reconstructed, up to isomorphism, from its field of functions.^{[lower-alpha 6]} In higher dimension the function field remembers less, but still decisive information about X. The study of function fields and their geometric meaning in higher dimensions is referred to as birational geometry. The minimal model program attempts to identify the simplest (in a certain precise sense) algebraic varieties with a prescribed function field.

Number theory: global fields

Global fields are in the limelight in algebraic number theory and arithmetic geometry. They are, by definition, number fields (finite extensions of Q) or function fields over F_q (finite extensions of F_q(t)). As for local fields, these two types of fields share several similar features, even though they are of characteristic 0 and positive characteristic, respectively. This function field analogy can help to shape mathematical expectations, often first by understanding questions about function fields, and later treating the number field case. The latter is often more difficult. For example, the Riemann hypothesis concerning the zeros of the Riemann zeta function (open as of 2017) can be regarded as being parallel to the Weil conjectures (proven in 1974 by Pierre Deligne).

The fifth roots of unity form a regular pentagon.

Cyclotomic fields are among the most intensely studied number fields. They are of the form Q(ζ_n), where ζ_n is a primitive nth root of unity, i.e., a complex number ζ that satisfies ζⁿ = 1 and ζ^m ≠ 1 for all 0 < m < n.^[57] For n being a regular prime, Kummer used cyclotomic fields to prove Fermat's Last Theorem, which asserts the non-existence of rational nonzero solutions to the equation

xⁿ + yⁿ = zⁿ.

Local fields are completions of global fields. Ostrowski's theorem asserts that the only completions of Q, a global field, are the local fields Q_p and R. Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally. This technique is called the local–global principle. For example, the Hasse–Minkowski theorem reduces the problem of finding rational solutions of quadratic equations to solving these equations in R and Q_p, whose solutions can easily be described.^[58]

Unlike for local fields, the Galois groups of global fields are not known. Inverse Galois theory studies the (unsolved) problem whether any finite group is the Galois group Gal(F/Q) for some number field F.^[59] Class field theory describes the abelian extensions, i.e., ones with abelian Galois group, or equivalently the abelianized Galois groups of global fields. A classical statement, the Kronecker–Weber theorem, describes the maximal abelian Q^ab extension of Q: it is the field

Q(ζ_n, n ≥ 2)

obtained by adjoining all primitive nth roots of unity. Kronecker's Jugendtraum asks for a similarly explicit description of F^ab of general number fields F. For imaginary quadratic fields, ${\displaystyle F=\mathbf {Q} ({\sqrt {-d}})}$, d > 0, the theory of complex multiplication describes F^ab using elliptic curves. For general number fields, no such explicit description is known.

Division rings

The hairy ball theorem states that a ball cannot be combed. More formally, there is no continuous tangent vector field on the sphere S², which is everywhere non-zero.

Dropping one or several axioms in the definition of a field leads to other algebraic structures. As was mentioned above, commutative rings satisfy all field axioms except for the existence of multiplicative inverses. Dropping instead commutativity of multiplication leads to the concept of a division ring or skew field;^{[lower-alpha 7]} sometimes associativity is weakened as well. The only division rings that are finite-dimensional R-vector spaces are R itself, C (which is a field), and the quaternions H (in which multiplication is non-commutative). This result is known as the Frobenius theorem. The octonions O, for which multiplication is neither commutative nor associative, is a normed alternative division algebra, but is not a division ring. This fact was proved using methods of algebraic topology in 1958 by Michel Kervaire, Raoul Bott, and John Milnor.^[62] The non-existence of an odd-dimensional division algebra is more classical. It can be deduced from the hairy ball theorem illustrated at the right.