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lds 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 field of fractions edit given an integral domain r its field of fractions q r is built with the fractions of two elements of r exactly as q is constructed from the integers more precisely the elements of q r are the fractions a b where a and b are in r and b 0 two fractions a b and c d are equal if and only if ad bc the operation on the fractions work exactly as for rational numbers for example a b c d a d b c b d displaystyle frac a b frac c d frac ad bc bd it is straightforward to show that if the ring is an integral domain the set of the fractions form a field 27 the field f x of the rational fractions over a field or an integral domain f is the field of fractions of the polynomial ring f x the field f x of formal laurent series i k a i x i k z a i f displaystyle sum _ i k infty a_ i x i k in mathbb z a_ i in f over a field f is the field of fractions of the ring f x of formal power series in which k 0 since any laurent series is a fraction of a power series divided by a power of x as opposed to an arbitrary power series the representation of fractions is less important in this situation though residue fields edit in addition to the field of fractions which embeds r injectively into a field a field can be obtained from a commutative ring r by means of a surjective map onto a field f any field obtained in this way is a quotient r m where m is a maximal ideal of r if r has only one maximal ideal m this field is called the residue field of r 28 the ideal generated by a single polynomial f in the polynomial ring r e x over a field e is maximal if and only if f is irreducible in e i e if f cannot be expressed as the product of two polynomials in e x of smaller degree this yields a field k e x f x this field k contains an element x namely the residue class of x which satisfies the equation f x 0 for example c is obtained from r by adjoining the imaginary unit symbol i which satisfies f i 0 where f x x 2 1 moreover f is irreducible over r which implies that the map that sends a polynomial f x r x to f i yields an isomorphism r x x 2 1 c displaystyle mathbf r x big left x 2 1 right stackrel cong longrightarrow mathbf c constructing fields within a bigger field edit 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 d field extensions edit further information glossary of field theory 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 f 4 f 2 are of degree 2 whereas r q is an infinite extension algebraic extensions edit a pivotal notion in the study of field extensions f e are algebraic elements an element x f is algebraic over e if it is a root of a polynomial with coefficients in e that is if it satisfies a polynomial equation e n x n e n 1 x n 1 e 1 x e 0 0 with e n e 0 in e and e n 0 for example the imaginary unit i in c is algebraic over r and even over q since it satisfies the equation i 2 1 0 a field extension in which every element of f is algebraic over e is called an algebraic extension any finite extension is necessarily algebraic as can be deduced from the above multiplicativity formula 31 the subfield e x generated by an element x as above is an algebraic extension of e if and only if x is an algebraic element that is to say if x is algebraic all other elements of e x are necessarily algebraic as well moreover the degree of the extension e x e i e the dimension of e x as an e vector space equals the minimal degree n such that there is a polynomial equation involving x as above if this degree is n then the elements of e x have the form k 0 n 1 a k x k a k e displaystyle sum _ k 0 n 1 a_ k x k a_ k in e for example the field q i of gaussian rationals is the subfield of c consisting of all numbers of the form a bi where both a and b are rational numbers summands of the form i 2 and similarly for higher exponents do not have to be considered here since a bi ci 2 can be simplified to a c bi transcendence bases edit the above mentioned field of rational fractions e x where x is an indeterminate is not an algebraic extension of e since there is no polynomial equation with coefficients in e whose zero is x elements such as x which are not algebraic are called transcendental informally speaking the indeterminate x and its powers do not interact with elements of e a similar construction can be carried out with a set of indeterminates instead of just one once again the field extension e x e discussed above is a key example if x is not algebraic i e x is not a root of a polynomial with coefficients in e then e x is isomorphic to e x this isomorphism is obtained by substituting x to x in rational fractions a subset s of a field f is a transcendence basis if it is algebraically independent do not satisfy any polynomial relations over e and if f is an algebraic extension of e s any field extension f e has a transcendence basis 32 thus field extensions can be split into ones of the form e s e purely transcendental extensions and algebraic extensions closure operations edit a field is algebraically closed if it does not have any strictly bigger algebraic extensions or equivalently if any polynomial equation f n x n f n 1 x n 1 f 1 x f 0 0 with coefficients f n f 0 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 x 2 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 it is rather special for the algebraic closure of some field f to be a finite extension of f because 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 q n 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 fields with additional structure edit since fields are ubiquitous in mathematics and beyond several refinements of the concept have been adapted to the needs of particular mathematical areas ordered fields edit main article ordered field 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 x 1 2 x 2 2 x n 2 0 displaystyle x_ 1 2 x_ 2 2 dots x_ n 2 0 has as its only solution x 1 x 2 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 edit 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 field q of the rational numbers in the sense that it is not in q but there are rational numbers that are arbitrarily close to it for the distance given by the absolute value the completion of q for the metric defined by the absolute value is the field of the real numbers r and the construction of the real numbers through cauchy sequences consists essentially of introducing new numbers for filling all such gaps field metric completion zero sequence q x y usual absolute value r 1 n q obtained using the p adic valuation for a prime number p q p p adic numbers p n f t f any field obtained using the t adic valuation f t t n 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 complex p adic numbers and is denoted c p 41 local fields edit the following topological fields are called local fields 42 e finite extensions of q p local fields of characteristic zero finite extensions of f p t the field of laurent series over f p local fields of characteristic p these two types of local fields share some fundamental similarities in this relation the elements p q p and t f p t referred to as the uniformizer correspond to each other the first manifestation of this is at an elementary level the elements of both fields can be expressed as power series in the uniformizer with coefficients in f p however since the addition in q p is done using carrying which is not the case in f p t these fields are not isomorphic the following facts show that this superficial similarity goes much deeper any first order statement that is true for almost all q p is also true for almost all f p t an application of this is the ax kochen theorem describing zeros of homogeneous polynomials in q p tamely ramified extensions of both fields are in bijection to one another adjoining arbitrary p power roots of p in q p respectively of t in f p t yields infinite extensions of these fields known as perfectoid fields strikingly the galois groups of these two fields are isomorphic which is the first glimpse of a remarkable parallel between these two fields 43 gal q p p 1 p gal f p t t 1 p displaystyle operatorname gal left mathbf q _ p bigl p 1 p infty bigr right cong operatorname gal left mathbf f _ p t bigl t 1 p infty bigr right differential fields edit differential fields are fields equipped with a derivation that is an operator d such that d a b d a b a d b displaystyle mathbf d ab mathbf d a b a mathbf d b for all elements a and b of the field 44 for example the field r x together with the standard derivative operator d on polynomials forms a differential field these fields are central to differential galois theory a variant of galois theory dealing with linear differential equations galois theory edit main article galois theory galois theory studies algebraic extensions of a field by studying the symmetry in the arithmetic operations of addition and multiplication an important notion in this area is that of finite galois extensions f e which are by definition those that are separable and normal the primitive element theorem shows that finite separable extensions are necessarily simple i e of the form f e x f x where f is an irreducible polynomial as above 45 for such an extension being normal and separable means that all zeros of f are contained in f and that f has only simple zeros the latter condition is always satisfied if e has characteristic 0 for a finite galois extension the galois group gal f e is the group of field automorphisms of f that are trivial on e i e the bijections σ f f that preserve addition and multiplication and that send elements of e to themselves the importance of this group stems from the fundamental theorem of galois theory which constructs an explicit one to one correspond...