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searching for The Rationals 231 found (253 total)

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{-5}{1}}} ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually

may or may not have a smallest element among the rationals. If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise

In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function. We start

{x^{3}}{3}}+xy^{2}+y^{2}+{\frac {1}{7}}=0} is a multivariate polynomial equation over the rationals. For many authors, the term algebraic equation refers only to the univariate

points on elliptic curves over the rationals. He showed that there are infinitely many elliptic curves over the rationals with the following torsion groups:

consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The

algebraically-closed field containing the rationals and so it is called the algebraic closure of the rationals. Any number that can be obtained from the

P=c(P)\operatorname {pp} (P).} This shows that every polynomial over the rationals is associated with a unique primitive polynomial over the integers,

over which it has degree two. Any number field that is Galois over the rationals must be either totally real or totally imaginary. Totally imaginary

numbers as an ordered monoid. The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in K {\displaystyle

and, more generally, CM fields. Any number field that is Galois over the rationals must be either totally real or totally imaginary. Section 13.1 of Alaca

In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from K× × K× to the group of nth roots of unity in a local field K such

continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. It also maps rational numbers to dyadic

essay Esquisse d'un Programme to study the absolute Galois group of the rationals by relating it to its action on the Teichmüller tower of Teichmüller

solution, then this also yields a real solution and a p-adic solution, as the rationals embed in the reals and p-adics: a global solution yields local solutions

basis for the algebraic integers in a simple algebraic extension of the rationals. Berwick was educated at a small private school before entering Bradford

of products of power-sum polynomials: they are a generating set over the rationals, but not over the integers. The power sum symmetric polynomial of degree

called irrational numbers. Some irrational numbers (as well as all the rationals) are the root of a polynomial with integer coefficients, such as the

quaternion algebra over the rationals ramifies is always even, and this is equivalent to the quadratic reciprocity law over the rationals. Moreover, the places

essentially the generalized Ramanujan conjecture for the group GL2 over the rationals at the infinite place, and says that the component at infinity of the

the first mathematician (1986) to show that some elliptic curves over the rationals have finite Tate–Shafarevich groups. It is widely believed that these

integers. These blocks are in turn densely ordered with the order type of the rationals. The result follows fairly easily because it is easy to see that the

{2}}} as a root; hence there is no linear or constant polynomial over the rationals having 2{\displaystyle {\sqrt {2}}} as a root. Furthermore, there is

correspond to: Roots of a given irreducible non-constant polynomial over the rationals with leading coefficient 1. For example, the type of square roots of

interval [ 0 , 1 ] {\displaystyle [0,1]} taking the value of one on the rationals, and zero, otherwise. Clearly the measure of this function should be

subset of the rationals. Including single-precision and double-precision IEEE 754 floats, among others Fixed-point representation of the rationals Integer

group is a group that is an extension of the absolute Galois group of the rationals by the Serre group. It was introduced by Langlands (1977) using an observation

instance, the real numbers form a resolvable topological space because the rationals and irrationals are disjoint dense subsets. A topological space that

familiar from arithmetic. Examples include the integers, the rationals and the real numbers. (The rationals and reals in fact form ordered fields.) The complex

In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology

functions. A differential field F is a field F0 (rational functions over the rationals Q for example) together with a derivation map u → ∂u. (Here ∂u is a

them. The open interval (0, 1) of rationals is order isomorphic to the rationals, since, for example, f ( x ) = 2 x − 1 1 − | 2 x − 1 | {\displaystyle

consequence of Gauss's lemma, this amounts to solving the problem also over the rationals. The algorithm starts by finding factorizations over suitable finite

problems related to existence and classification of Abelian varieties over the rationals with bad reduction in one prime only, and algorithms. In the past, René

leading coefficient is positive, The polynomial is irreducible over the rationals (and integers), and There is no common factor for all the infinitely

given norm in an ideal class of a definite quaternion algebra over the rationals, and that give a representation of the Hecke algebra. Eichler (1955)

representations correspond to CM-motives over the algebraic closure of the rationals, or to polarizable rational Hodge structures with abelian Mumford–Tate

Euclidean closure of the rational numbers, the smallest field extension of the rationals that includes the square roots of all of its positive numbers. The proof

free Lie algebra on a countable set of generators over the rationals. Over the rationals this identifies the Hopf algebra NSYmm with the universal enveloping

quadratic extension. The converse holds too. (An example is K equal to the rationals and L equal to an imaginary quadratic field; both have unit rank 0.)

Tages 2. The song appeared as "Leavin' Here" on singles released by the Rationals in 1966 and 1967, as did a 1989 version by Tinsley Ellis on his debut

In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the

|⋅|{\displaystyle |\cdot |} and |⋅|∗{\displaystyle |\cdot |_{*}} on the rationals are defined to be equivalent if they induce the same topology; this

problem for equations over the rationals remains open. Barry Mazur has conjectured that for any variety over the rationals, the topological closure over

algebraic, irrational, and 1, b1, ..., bn are linearly independent over the rationals, and the ai are all algebraic and not 0 or 1. Baker (1977) also gave

group over a cyclotomic extension of the rationals. (More precisely, over the cyclotomic extension of the rationals generated by the values of the irreducible

including the MC5, Parliament-Funkadelic, the Ohio Players, In early '69, the Rationals began sessions for their first LP the Detroit Emeralds, the Fantastic

where K {\displaystyle K} is the finite collection of rationals. Since the rationals are countable, we can look at the pointwise limit of these things over

David W. (July 1995). "LCF: A Lexicographic Binary Representation of the Rationals". J.UCS the Journal of Universal Computer Science. Vol. 1. pp. 484–503

irreducible over the rationals if and only if it is irreducible over the integers. This implies also that the factorization over the rationals of a polynomial

1016/S0304-3975(96)00101-6 Calkin, Neil; Wilf, Herbert (2000), "Recounting the rationals" (PDF), American Mathematical Monthly, 107 (4), Mathematical Association

computer to check that it vanishes for the cyclotomic Zp-extension of the rationals for all primes less than 4000. Iwasawa (1971) later conjectured that

arbitrary number fields and more general cubic equations. For curves over the rationals, the generalization says that, for a nonsingular cubic curve whose Weierstrass

Aronszajn tree is called special if there is a function f from the tree to the rationals so that f(x) < f(y) whenever x < y. Martin's axiom MA( ℵ 1 {\displaystyle

to the rationals (as for any other field of characteristic 0), and the order on this rational subfield is the same as the order of the rationals themselves

complex cobordism. Complex cobordism over the rationals can be reduced to ordinary cohomology over the rationals, so the main interest is in the torsion

w(2−i)=0{\displaystyle w(2-i)=0}) is formally 5-adic (the place v=5 of the rationals splits in two places of the Gaussian rationals since X2+1{\displaystyle

smaller field, it is sufficient that f is an irreducible polynomial over the rationals. Similarly, one may define the ring of integers O Q [ r ] {\textstyle

Pythagorean sect. This forced a distinction between numbers (integers and the rationals—the subjects of arithmetic), on the one hand, and lengths and proportions

the set of natural numbers since the rationals are countable: there is a bijection from the naturals to the rationals. List of paradoxes – List of statements

takes values in the m roots of 1. When m = 2 and the global field is the rationals this is more or less the same as the Jacobi symbol. Hilbert symbol The

5519, 9419, 9539, 9929,... OEIS: A006962 If an elliptic curve over the rationals has complex multiplication then the set of primes for which it is supersingular

universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal

two ways: they eventually include all positive rationals, not just the rationals within the interval [0,1], and at the nth step all mediants are included

French), 1: 681–704, JFM 35.0387.02 See, e.g., Mihăilă, Ioana (2004), "The rationals of the Cantor set", The College Mathematics Journal, 35 (4): 251–255

imply gaplessness: the rationals do not enjoy the least upper bound property. That means that if one were to partition the rationals into two non-empty

algebraic integers. The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object

say that both a set and its complement exhibit typical behavior, both the rationals and irrationals cannot be examples of sets large enough to be typical

and the rationals, each with the metric induced from R {\displaystyle \mathbb {R} } . One can think of (0, 1) as "missing" its endpoints 0 and 1. The rationals

Victor A. Kolyvagin extended this to modular elliptic curves over the rationals of analytic rank at most 1 (The modularity theorem later showed that

For instance, over the integers, the associated symmetric form (over the rationals) may take half-integer values, while over Z / 2 Z , {\displaystyle \mathbb

the quadratic algebraic numbers and the rational numbers, and between the rationals and the dyadic rationals. Any binary relation R is said to be dense

ISSN 0895-4801. Calkin, Neil; Wilf, Herbert S. (2000). "Recounting the Rationals". The American Mathematical Monthly. 107 (4). Informa UK Limited: 360–363

in topology, the spectral sequence degenerates after tensoring with the rationals. For arbitrary schemes of finite type over a field (not necessarily

expressed as saying that if log α, log γ are linearly independent over the rationals, then they are linearly independent over the algebraic numbers. The

Expressed as bit strings, the periodic orbits of the map can be seen to the rationals. That is, after an initial "chaotic" sequence of b 0 , b 1 , b 2 , …

y2 = x3 − x and y2 = x3 − x + 1. If we consider these as curves over the rationals, then the modularity theorem asserts that they can be parametrized by

of Q ( ω ) {\displaystyle \mathbb {Q} (\omega )} over the field of the rationals. The rules of exponentiation imply that the composition of two such

ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as their direct summands, the Prüfer groups

Curves of genus > 1 over the rationals have only finitely many rational points

Diophantine equation involving an irreducible bivariate form of deg > 2 over the rationals Thue–Morse sequence – Infinite binary sequence generated by repeated

the ancient Greeks. In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example

function f : Q → R {\displaystyle f:\mathbb {Q} \to \mathbb {R} } from the rationals Q {\displaystyle \mathbb {Q} } to R , {\displaystyle \mathbb {R} ,}

numbers between 0 and 1, given that the mass of the interval [0, 1] is 1. The rationals are dense in the reals, so any value between and including 0 and 1 may

otherwise. Minkowski's question mark function: Derivatives vanish on the rationals. Weierstrass function: is an example of continuous function that is

even be of the same cardinality. Perhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need

countable dense total order without endpoints, namely the order type of the rationals. Tarski-Vaught test Chang, Chen Chung; Keisler, H. Jerome (1990) [1973]

curve. However a quaternion algebra over the rationals cannot act on a 2-dimensional vector space over the rationals. The same argument eliminates the possibility

such as the integers and the rationals, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication

As an example of a cogenerator in the same category, we have Q/Z, the rationals modulo the integers, which is a divisible abelian group. Given any abelian

torsion-free abelian groups (essentially the same as vector spaces over the rationals). The theory of the set of natural numbers with a successor function

plane One point compactification topology One point compactification of the rationals Hilbert space Fréchet space Hilbert cube Order topology Open ordinal

injective: consider, for example, the quotient map q : Q → Q/Z, where Q is the rationals under addition, Z the integers (also considered a group under addition)

sometimes omitted, especially if it is Z). Usually G is the integers, the rationals, the reals, the complex numbers, or the integers mod a prime p. The

numbers do not – for example, x3 − x = x(x − 1)(x + 1) factors over the rationals, but its derivative, 3 x 2 − 1 = 3 ( x − 1 3 ) ( x + 1 3 ) , {\displaystyle

Suppose K is a number field (a finite-dimensional field extension of the rationals Q) with ring of integers OK (this ring is the integral closure of the

{\displaystyle k} points. This can also be proven directly for the ordering on the rationals, by constructing a piecewise linear order automorphism with breakpoints

performs sieving in two different fields. The first field is usually the rationals. The second is a higher degree field. The efficiency of the algorithm

using the Lebesgue measure: Indeed, it is the indicator function of the rationals so by definition ∫ [ 0 , 1 ] 1 Q d μ = μ ( Q ∩ [ 0 , 1 ] ) = 0 , {\displaystyle

noncommutative number theory as generalizations of number fields (extensions of the rationals Q); see noncommutative number field. Azumaya algebra, generalization

constructed using quaternion algebras associated with quadratic extensions of the rationals by class field theory. In this case Selberg's trace formula shows that

of the rationals, other than the identity and complex conjugation, do not usually act continuously on a complex variety defined over the rationals, so

density given by the order of the pole at s=1, which is 1 (When K is the rationals, this is essentially Euler's proof that there are infinitely many primes

the space of real numbers. The above examples, the irrationals and the rationals, are also dense sets in their topological space, namely R {\displaystyle

{\displaystyle X} and Y {\displaystyle Y} are also a unital modules over the rationals Q {\displaystyle \mathbb {Q} } (such as real or complex vector spaces)

\mathbb {R} ^{2}} consisting of the open upper half plane together with the rationals on the x-axis, namely, X = ( R × ( 0 , ∞ ) ) ∪ ( Q × { 0 } ) , {\displaystyle

corresponding ring of Witt vectors. It cannot in general be realized over the rationals or over the local field Qp, suggesting that there is no easy way to

projective variety X, the Hodge conjecture predicts the image (tensored with the rationals Q) of the cycle map from the Chow groups to singular cohomology. For

L=\mathbf {Q} (\zeta _{p^{n}})/\mathbf {Q} } be a cyclotomic extension of the rationals. The Galois group G {\displaystyle G} equals ( Z / p n ) × {\displaystyle

K.L\cong K\otimes _{N}L} , as with the aforementioned extensions of the rationals. A significant case in the theory of cyclotomic fields is that for the

Indefinite quadratic form in > 4 variables over the rationals nontrivially represents 0

subring is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of

x ) {\displaystyle \left(a/q-cr_{0}/qx,a/q+cr_{0}/qx\right)} around the rationals a / q , q < r 0 {\displaystyle a/q,q<r_{0}} where c {\displaystyle c}

JSTOR 1970860. Baumgartner, J., J. Malitz, and W. Reiehart, Embedding trees in the rationals, Proc. Natl. Acad. Sci. U.S.A., 67, pp. 1746 – 1753, 1970 Shelah, S

simplifications (the real numbers have properties very different from the rationals, and likewise continuous maps have very different properties from smooth

numbers, this corresponds to the fact that an abelian extension of the rationals lying in one cyclotomic field also lies in infinitely many other cyclotomic

elements of PSL(2,Z) will carry z = 0 to every rational number, and the rationals Q are dense in R. A linear fractional transformation defined by a matrix

Sky (which featured a young Doug Fieger), and blue-eyed soul rockers the Rationals. In 1965 Mitch Ryder & the Detroit Wheels had a national top 10 hit

the ergodic perspective (linking the p-adic and regular norms over the rationals) by showing that the Artin–Hasse exponential is also the generating

the same. For example, consider the subset Y = {−1} ∪ {1/n }n∈N of the rationals. Under the subspace topology, the singleton set {−1} is open in Y, but

George Clinton, Bob Seger and the Silver Bullet Band, Mitch Ryder, and The Rationals. Madonna was a dance major at the University of Michigan in the late

irrational numbers as the least upper bounds of certain subsets of the rationals. The least-upper-bound property is equivalent to other forms of the

which is the dimension of interest in critical superstring theory. Over the rationals Michael Atiyah and Graeme Segal have shown that all of the differentials

common divisors and factorization for multivariate polynomials over the rationals, finite fields, algebraic number fields, and algebraic function fields

are periodic. Let β > 1, and Q(β) be the smallest field extension of the rationals containing β. Then any real number in [0,1) having a periodic β-expansion

f(x+y)=f(x)f(y)} can be constructed using a basis for the real numbers over the rationals, as described by Hewitt and Stromberg. Elementary definition by powers

fail. If random bits are in short supply, the Chen-Kao algorithm (over the rationals) or the Lewin-Vadhan algorithm (over any field) require fewer random

non-zero algebraic number; then {α} is a linearly independent set over the rationals, and therefore by the first formulation of the theorem {eα} is an algebraically

L-functions) that extend the notion of a Dirichlet L-function from the rationals to other number fields. For a Hecke character χ, its L-function is defined

independently formulated the torsion conjecture for elliptic curves over the rationals after it was originally formulated by Beppo Levi in 1908. Nagell was

Q / Z {\displaystyle \mathbb {Q} /\mathbb {Z} } is the quotient of the rationals by the integers, taken as an additive group. Notice that in the torsion

Pearl Jam, The Birds, The Who, Brownsville Station, The Messengers, The Rationals, and The Volts "Locking Up My Heart" The Marvelettes "What Goes Up Must

furthermore bijectively maps the projectively extended rational line (the rationals with infinity) to itself, the irrationals to the irrationals, the transcendental

differentially closed field of characteristic 0 (the differential closure of the rationals) is not minimal; this was a rather surprising result, as it is not what

rational numbers and v runs over all the non-trivial absolute values of the rationals (the Archimedean one and the p-adic absolute values for primes p). The

differentially closed field of characteristic 0 (the differential closure of the rationals) is not minimal; this was a rather surprising result, as it is not what

demonstrating that elementary number theory could be defined in terms of the rationals. (Elementary number theory was already known to be undecidable by Gödel's

of the hyperreals, and they have the same first-order properties as the rationals. Consider, however, the Archimedean property of the reals, which states

rational numbers and v runs over all the non-trivial absolute values of the rationals (the Archimedean one and the p-adic absolute values for primes p). The

countable set Z × N {\displaystyle \mathbb {Z} \times \mathbb {N} } to the rationals Q {\displaystyle \mathbb {Q} } . Proof: Per definition, every algebraic

rationals in [0, 1]." Cantor lets φν denote a sequence enumerating the rationals in [0, 1], which is the kind of sequence needed for his construction

numbers have rank one, as well as every nonzero additive subgroup of the rationals. On the other hand, the multiplicative group of the nonzero rationals

number in the usual sense. When K {\displaystyle K} is larger than the rationals it is easy to write down Carmichael ideals in O K {\displaystyle {\mathcal

Tutte's characterization of binary matroids that are representable over the rationals (Whittle 1995), perhaps the biggest single contribution of the 1990s

transition from quantum to classical is interpreted as tensoring by the rationals. So the F's must be some rational version of twisted K-theory. Such

has conjectured the existence of an absolute constant A, such that the rationals with partial quotients restricted by A contain at least one for every

\mathbb {Z} } is nowhere dense in R {\displaystyle \mathbb {R} } but the rationals Q {\displaystyle \mathbb {Q} } are not (they are dense everywhere).

Diophantine equation involving an irreducible bivariate form of deg > 2 over the rationals

are Cauchy sequences of rational numbers that are not convergent in the rationals, e.g. the sequence defined by x 1 = 1 {\displaystyle x_{1}=1} and x

the supremum of a set of rationals is irrational, which means that the rationals are incomplete. One basic property of the supremum is sup { f ( t )

linear program computes the "fractional clique number", a relaxation to the rationals of the integer concept of clique number. That is, a weighting of the

The Perles configuration, linear over the reals but not the rationals

"Tele-Gib" which he had constructed as a gift to Beck. Scott Morgan of the Rationals, who at one point shared a dressing room with the Yardbirds, recalls

counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively

modular, which extends results #2 and #3 to all elliptic curves over the rationals, and shows that the L-functions of all elliptic curves over Q are defined

work. An algebraic basis for the real numbers as a vector space over the rationals is known as a Hamel basis (note that some authors use this term in a

the realization of every finite solvable group as a Galois group over the rationals. Another development is the Golod–Shafarevich theorem on towers of unramified

exponent of the conductor. The conductor of an elliptic curve over the rationals was introduced and named by Weil (1967) as a constant appearing in the

prime numbers p, and the corresponding maximal abelian extension of the rationals is the field generated by all roots of unity. This is known as the Kronecker–Weber

x} can be named by giving an infinite set of rationals, namely all the rationals less than x {\displaystyle x} . We will in effect define x {\displaystyle

but the other three polynomials are irreducible (as a polynomial over the rationals, 3 is a unit, and, therefore, does not count as a factor). Over the

integer, if one starts with the lowest integer as first approximant, the rationals generated all appear in the list of convergents for the continued fraction

{1}{n}}\right]{\text{ and }}\left[{\frac {1}{\pi }}+{\frac {1}{n}},1\right]} cover all the rationals in [0, 1] for n = 4, 5, ...  but this cover does not have a finite subcover

irreducible representations of the Galois group of a solvable extension of the rationals are automorphic as predicted by the Langlands conjectures. The functions

many sub-semirings of R {\displaystyle {\mathbb {R} }} , including the rationals, the integers, as well as the non-negative parts of each of these structures

equations in the integers Z {\displaystyle \mathbb {Z} } rather than the rationals Q . {\displaystyle \mathbb {Q} .} For homogeneous polynomial equations

discontinuous at every irrational point. The indicator function of the rationals, also known as the Dirichlet function, is discontinuous everywhere.

curve of degree higher than two has no point in the affine plane over the rationals." Geometry in affine spaces over a finite fields has also been widely

the true inverse. Any totally ordered group under addition such as the rationals or the reals can be substituted for the integers in this example. The

modulo a prime is isomorphic to a subgroup of the Galois group over the rationals. A permutation group on 5 objects with elements of orders 6 and 5 must

from division. Later topics in this part include the countability of the rationals, the irrationality of the square root of 2, exponentiation and logarithms

each sex to a mating group called a "triad," and acting as mentors to the Rationals. Little is shown of "hard one" society; whereas Dua suspects that the

rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually

Jean-Marc Fontaine shows that any positive-dimensional abelian variety over the rationals has bad reduction somewhere. 2001 Proof of the modularity theorem for

taking square roots. This follows because its minimal polynomial over the rationals has degree 3. This construction is possible using a straightedge with

numbers, ζ be the order type of the integers, and η be the order type of the rationals, the order type of any countable nonstandard model of PA is ω + ζ·η

obtains R or C, if it is non-Archimedean and lies over a prime p of the rationals, one obtains a finite extension K w / Q p : {\displaystyle K_{w}/\mathbf

rational number p/q. It is sometimes said that the circle map maps the rationals, a set of measure zero at K = 0, to a set of non-zero measure for K ≠ 0

a−1 ∗ b, and y ∗ a = b if and only if y = b ∗ a−1. The integers Z (or the rationals Q or the reals R) with subtraction (−) form a quasigroup. These quasigroups

"Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals". Automated Deduction – CADE-24. Lecture Notes in Computer Science.

has a maximal ideal.theorem III.5.5 Every countable vector space over the rationals (or over any countable field) has a basis.theorem III.4.3 For any countable

function on the reals is uniquely determined by its restriction to the rationals.) The set C[a, b] of continuous real-valued functions on [a, b], together

example e = 5; s = 1.234571 and e = 5; s = 1.234567 are approximations to the rationals 123457.1467 and 123456.659. e=5; s=1.234571 − e=5; s=1.234567 ----------------

terms of radicals) for the solutions of general quintic equations over the rationals; this statement is known as the Abel–Ruffini theorem, first asserted

quotient field is an algebraic number field (a finite extension of the rationals) then Milnor (1971, corollary 16.3) shows that SK1(A) vanishes. The

an nth power everywhere locally but not globally. The element 16 in the rationals is an 8th power at all places except 2, but is not an 8th power in the

groups under addition where the identity is typically 0. Note that with the rationals, zero must be excluded because, under multiplication, it does not have

indeterminates. The reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general

However, if the entries of A are all algebraic numbers, which include the rationals, the eigenvalues must also be algebraic numbers (that is, they cannot

they are equal. (Indeed, this works with any integer-base system.) The rationals thus non-uniquely expressible are those of form z ( r + 1 ) + 1 b i

axiomatizes the real number line by some means, a collection extending the rationals and with an apartness relation definable from an asymmetric order structure

paramodular group is a subgroup of the usual symplectic group (over the rationals) but its coordinates are not always integers. These two forms of the

follow that the real numbers, being the union of the irrationals and the rationals (which, as a countable set, is evidently meager), would also be a meager

subtraction, and multiplication are all primitive recursive. Similarly, if the rationals are represented by Gödel numbers then the field operations are all primitive

Mod⁡(Q){\displaystyle \operatorname {Mod} ({\mathbb {Q}})} of all vector spaces over the rationals, with the canonical functor Ab→Mod⁡(Q){\displaystyle \mathbf {Ab} \to

{\displaystyle {\mathfrak {n}}} of N {\displaystyle N} can be defined over the rationals. That is, if and only if the structure constants of n {\displaystyle

This construction is parallel to the construction of the reals from the rationals given by Cantor. He started with the ring of the Cauchy sequences of

Φ {\displaystyle \Phi } . A theorem of René Thom, which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators

linked to totally ramified primes, as follows: if a field extension of the rationals is generated by the root of a polynomial that is Eisenstein at p then

ISSN 0002-9939. Widmer, Martin (2011). "On certain infinite extensions of the rationals with Northcott property". Monatshefte für Mathematik. 162 (3): 341–353

log(1 + x + y + xy) = log(1 + x) + log(1 + y). If R does not contain the rationals, a map f can be constructed by extension of scalars to R ⊗ Q, but this

example e = 5; s = 1.234571 and e = 5; s = 1.234567 are approximations to the rationals 123457.1467 and 123456.659. e=5; s=1.234571 − e=5; s=1.234567 ----------------

the extending map h in the above definition is typically not unique. The rationals Q (with addition) form an injective abelian group (i.e. an injective

the p {\displaystyle p} -adic numbers may similarly play the role of the rationals; in particular, we can define the norm and trace in exactly the same

For example, the minimal polynomial (over the reals as well as over the rationals) of the complex number i is X 2 + 1 {\displaystyle X^{2}+1} . The cyclotomic

but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable. He also showed that all countable

linear group over the rational adeles by the special linear group of the rationals (a compact set, because S L 2 ( Q ) {\displaystyle SL_{2}(\mathbb {Q}

number fields. The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis

finite. The class field extensions of finite transcendence degree over the rationals. This theorem, named after Nikolai Luzin and his doctoral student Pyotr

i {\displaystyle i} th prime number in the multiplicative group of the rationals as instead giving the coefficient of x i − 1 {\displaystyle x^{i-1}}

\ldots ,{\frac {\ln p_{N}}{2\pi }}} (which are linearly independent over the rationals) we can find real values of t so that ln ⁡ ( ζ M ( s , θ ^ ) ) {\displaystyle

the family is less than ε on each open set in the cover. The role of the rationals can then be played by a set of points drawn from each open set in each

} where α Q {\displaystyle \alpha \mathbb {Q} } is a scaled copy of the rationals. This shows that only linear solutions are permitted when the domain

additive in x . {\displaystyle x.} It follows that it is linear over the rationals, thus linear by continuity. Several characterizations of spaces isomorphic

over finite fields obtained from reducing a given elliptic curve over the rationals. Mikio Sato and, independently, John Tate suggested it around 1960.

is the order of the ideal class group of an algebraic extension of the rationals with discriminant n. The notation is ambiguous, as there are in general

order 47 (although this group can be realized as a Galois group over the rationals in other ways). The inverse Galois problem remains unsolved. Noether

A tender salutation, or, The substance of a letter given forth by the Rationals, to the despised remnant and seed of God, in the people called Quakers

Sullivan algebra is a commutative differential graded algebra over the rationals Q {\displaystyle \mathbb {Q} } , whose underlying algebra is the free

from 2 and with A ≠ ±2 and B ≠ 0, but they are also considered over the rationals with the same restrictions for A and B. It is possible to do some "operations"

unit of the ring. The extra points can be associated with Q ⊂ R ⊂ C, the rationals in the extended complex upper-half plane. The group of homographies

Bunce–Deddens algebra is precisely the corresponding dense subgroup of the rationals Q. As it follows from the definition that two Bunce–Deddens algebras

problem: is every finite group the Galois group of a Galois extension of the rationals? Are there an infinite number of Leinster groups? Does generalized moonshine

case the signs of the remainders coincide with those computed over the rationals. Note that the algorithm for computing the subresultant pseudo-remainder

published a proof that every finite solvable group is a Galois group over the rationals. However Schmidt[who?] pointed out a gap in the argument at the prime

}I_{n}} could yield ∅ {\displaystyle \emptyset } , if only considering the rationals. The axiom is equivalent to the existence of the infimum and supremum

may be seen as but one of infinitely many different completions of the rationals, with no logical reason to favour it over the various p-adic completions

a power of 2, making f ( x ) {\displaystyle f(x)} irreducible over the rationals. (This particular f ( x ) {\displaystyle f(x)} comes from the family

Theorem (Wehrung 1998). There exists a dimension vector space G over the rationals with order-unit whose positive cone G+ is not isomorphic to V(R), for

stands for additive idele. Thus, an adele is an additive ideal element. The rationals K = Q {\displaystyle K={\mathbf {Q}}} have a valuation for every prime

special A special Aronszajn tree is one with an order preserving map to the rationals square The square principle is a combinatorial principle holding in