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The history of each subfield is briefly addressed in its own section below; see the main article of each subfield for fuller treatments. Many of the most interesting questions in each area remain open and are being actively worked on.-->
==SubdivisiBagian divisi utama==
===Teori bilangan dasar===
- Dalam pengembangan -
Istilah ''[[bukti dasar|elemen dasar]]'' biasanya menampakkan metode yang bukan menggunakan [[analisis kompleks]]. Misalnya, [[teorema bilangan prima]] pertama kali dibuktikan menggunakan analisis kompleks pada tahun 1896, tetapi bukti dasar baru ditemukan pada tahun 1949 oleh [[Paul Erdős|Erdős]] dan [[Atle Selberg|Selberg]].{{sfn|Goldfeld|2003}} Istilah ini sedikit ambigu: misal, bukti berdasarkan [[teorema Tauberian]] kompleks (misalnya, [[teorema Wiener–Ikehara|Wiener–Ikehara]]) merupakan pencerahan yang tidak cukup mendasar meskipun menggunakan [[analisis Fourier]], dibandingkan analisis kompleks seperti itu. Ini seperti penempatan berbeda, bukti "dasar" mungkin lebih panjang dan lebih sulit bagi sebagian besar pembaca dibanding bukti non-dasar.
<!--[***]===Elementary tools===
The term ''[[elementary proof|elementary]]'' generally denotes a method that does not use [[complex analysis]]. For example, the [[prime number theorem]] was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by [[Paul Erdős|Erdős]] and [[Atle Selberg|Selberg]].{{sfn|Goldfeld|2003}} The term is somewhat ambiguous: for example, proofs based on complex [[Tauberian theorem]]s (for example, [[Wiener–Ikehara theorem|Wiener–Ikehara]]) are often seen as quite enlightening but not elementary, in spite of using Fourier analysis, rather than complex analysis as such. Here as elsewhere, an ''elementary'' proof may be longer and more difficult for most readers than a non-elementary one.
[[Berkas:Paul Erdos with Terence Tao.jpg|thumb|270px|Number theorists [[Paul Erdős]] and [[Terence Tao]] in 1985, when Erdős was 72 and Tao was 10.]] ▼Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics.<ref>See, for example, the initial comment in {{harvnb|Iwaniec|Kowalski|2004|p=1}}.</ref>
▲[[Berkas:Paul Erdos with Terence Tao.jpg|thumb|270px| NumberAhli theoriststeori bilangan [[Paul Erdős]] anddan [[Terence Tao]] inpada tahun 1985, whenketika Erdős wasberusia 72 andtahun dan Tao wasberusia 10 tahun.]]
===Analytic number theory===
{{main|Analytic number theory}}
Teori bilangan memiliki reputasi sebagai bidang yang banyak hasilnya pula bisa dinyatakan kepada orang awam. Pada saat yang sama, bukti dari hasil ini tidak dapat diakses secara khusus, sebagian karena jangkauan alat yang mereka gunakan, jika ada maka sangat luas dalam matematika.<ref>Lihat, contohnya, di komentar awal {{harvnb|Iwaniec|Kowalski|2004|p=1}}.</ref>
[[Gambar:Complex zeta.jpg|right|thumb|[[Riemann zeta function]] ζ(''s'') in the [[complex plane]]. The color of a point ''s'' gives the value of ζ(''s''): dark colors denote values close to zero and hue gives the value's [[Argument (complex analysis)|argument]].]]
[[Berkas:ModularGroup-FundamentalDomain.svg|thumb|The action of the [[modular group]] on the [[upper half plane]]. The region in grey is the standard [[fundamental domain]].]]
''Analytic number theory'' may be defined
* in terms of its tools, as the study of the integers by means of tools from real and complex analysis;{{sfn|Apostol|1976|p=7}} or
* in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities.<ref>{{harvnb|Granville|2008|loc=section 1}}: "The main difference is that in algebraic number theory [...] one typically considers questions with answers that are given by exact formulas, whereas in analytic number theory [...] one looks for ''good approximations''."</ref>
Some subjects generally considered to be part of analytic number theory, for example, [[sieve theory]],<ref group="note">Sieve theory figures as one of the main subareas of analytic number theory in many standard treatments; see, for instance, {{harvnb|Iwaniec|Kowalski|2004}} or {{harvnb|Montgomery|Vaughan|2007}}</ref> are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis,<ref group="note">This is the case for small sieves (in particular, some combinatorial sieves such as the [[Brun sieve]]) rather than for [[large sieve]]s; the study of the latter now includes ideas from [[harmonic analysis|harmonic]] and [[functional analysis]].</ref> yet it does belong to analytic number theory.
The following are examples of problems in analytic number theory: the [[prime number theorem]], the [[Goldbach conjecture]] (or the [[twin prime conjecture]], or the [[Hardy–Littlewood conjecture]]s), the [[Waring problem]] and the [[Riemann hypothesis]]. Some of the most important tools of analytic number theory are the [[circle method]], [[sieve theory|sieve methods]] and [[L-functions]] (or, rather, the study of their properties). The theory of [[modular form]]s (and, more generally, [[automorphic forms]]) also occupies an increasingly central place in the toolbox of analytic number theory.<ref>See the remarks in the introduction to {{harvnb|Iwaniec|Kowalski|2004|p=1}}: "However much stronger...".</ref>
One may ask analytic questions about [[algebraic number]]s, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define [[prime ideals]] (generalizations of [[prime number]]s in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question [[Landau prime ideal theorem|can be answered]] by means of an examination of [[Dedekind zeta function]]s, which are generalizations of the [[Riemann zeta function]], a key analytic object at the roots of the subject.<ref>{{harvnb|Granville|2008|loc=section 3}}: "[Riemann] defined what we now call the Riemann zeta function [...] Riemann's deep work gave birth to our subject [...]"</ref> This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function.<ref>See, for example, {{harvnb|Montgomery|Vaughan|2007}}, p. 1.</ref>
===Algebraic number theory===
{{main|Algebraic number theory}}
An ''[[algebraic number]]'' is any complex number that is a solution to some polynomial equation <math>f(x)=0</math> with rational coefficients; for example, every solution <math>x</math> of <math>x^5 + (11/2) x^3 - 7 x^2 + 9 = 0 </math> (say) is an algebraic number. Fields of algebraic numbers are also called ''[[algebraic number field]]s'', or shortly ''[[number field]]s''. Algebraic number theory studies algebraic number fields.{{sfn|Milne|2017|p=2}} Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study.
It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as the discussion of quadratic forms in ''Disquisitiones arithmeticae'' can be restated in terms of [[ideal (ring theory)|ideals]] and
[[Norm (mathematics)|norms]] in quadratic fields. (A ''quadratic field'' consists of all
numbers of the form <math> a + b \sqrt{d}</math>, where
<math>a</math> and <math>b</math> are rational numbers and <math>d</math>
is a fixed rational number whose square root is not rational.)
For that matter, the 11th-century [[chakravala method]] amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither [[Bhāskara II|Bhāskara]] nor Gauss knew of number fields as such.
The grounds of the subject as we know it were set in the late nineteenth century, when ''ideal numbers'', the ''theory of ideals'' and ''valuation theory'' were developed; these are three complementary ways of dealing with the lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationals
and <math> \sqrt{-5}</math>, the number <math>6</math> can be factorised both as <math> 6 = 2 \cdot 3</math> and
<math> 6 = (1 + \sqrt{-5}) ( 1 - \sqrt{-5})</math>; all of <math>2</math>, <math>3</math>, <math>1 + \sqrt{-5}</math> and
<math> 1 - \sqrt{-5}</math>
are irreducible, and thus, in a naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by [[Ernst Kummer|Kummer]]) seems to have come from the study of higher reciprocity laws,{{sfn|Edwards|2000|p=79}} that is, generalisations of [[quadratic reciprocity]].
Number fields are often studied as extensions of smaller number fields: a field ''L'' is said to be an ''extension'' of a field ''K'' if ''L'' contains ''K''.
(For example, the complex numbers ''C'' are an extension of the reals ''R'', and the reals ''R'' are an extension of the rationals ''Q''.)
Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions ''L'' of ''K'' such that the [[Galois group]]<ref group="note">The Galois group of an extension ''L/K'' consists of the operations ([[isomorphisms]]) that send elements of L to other elements of L while leaving all elements of K fixed.
Thus, for instance, ''Gal(C/R)'' consists of two elements: the identity element
(taking every element ''x'' + ''iy'' of ''C'' to itself) and complex conjugation
(the map taking each element ''x'' + ''iy'' to ''x'' − ''iy'').
The Galois group of an extension tells us many of its crucial properties. The study of Galois groups started with [[Évariste Galois]]; in modern language, the main outcome of his work is that an equation ''f''(''x'') = 0 can be solved by radicals
(that is, ''x'' can be expressed in terms of the four basic operations together
with square roots, cubic roots, etc.) if and only if the extension of the rationals by the roots of the equation ''f''(''x'') = 0 has a Galois group that is [[solvable group|solvable]]
in the sense of group theory. ("Solvable", in the sense of group theory, is a simple property that can be checked easily for finite groups.)</ref> Gal(''L''/''K'') of ''L'' over ''K'' is an [[abelian group]]—are relatively well understood.
Their classification was the object of the programme of [[class field theory]], which was initiated in the late 19th century (partly by [[Leopold Kronecker|Kronecker]] and [[Gotthold Eisenstein|Eisenstein]]) and carried out largely in 1900–1950.
An example of an active area of research in algebraic number theory is [[Iwasawa theory]]. The [[Langlands program]], one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields.
===Diophantine geometry===
{{main|Diophantine geometry}}
The central problem of ''Diophantine geometry'' is to determine when a [[Diophantine equation]] has solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object.
For example, an equation in two variables defines a curve in the plane. More generally, an equation, or system of equations, in two or more variables defines a [[algebraic curve|curve]], a [[algebraic surface|surface]] or some other such object in ''n''-dimensional space. In Diophantine geometry, one asks whether there are any ''[[rational points]]'' (points all of whose coordinates are rationals) or
''integral points'' (points all of whose coordinates are integers) on the curve or surface. If there are any such points, the next step is to ask how many there are and how they are distributed. A basic question in this direction is if there are finitely
or infinitely many rational points on a given curve (or surface).
In the [[Pythagorean theorem|Pythagorean equation]] <math>x^2+y^2 = 1,</math>
we would like to study its rational solutions, that is, its solutions
<math>(x,y)</math> such that
''x'' and ''y'' are both rational. This is the same as asking for all integer solutions
to <math>a^2 + b^2 = c^2</math>; any solution to the latter equation gives
us a solution <math>x = a/c</math>, <math>y = b/c</math> to the former. It is also the
same as asking for all points with rational coordinates on the curve
described by <math>x^2 + y^2 = 1</math>. (This curve happens to be a circle of radius 1 around the origin.)
[[Gambar:ECClines-3.svg|right|thumb|300px|Two examples of an [[elliptic curve]], that is, a curve
of genus 1 having at least one rational point. (Either graph can be seen as a slice of a [[torus]] in four-dimensional space.)]]
The rephrasing of questions on equations in terms of points on curves turns out to be felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve—that is, rational or integer solutions to an equation <math>f(x,y)=0</math>, where <math>f</math> is a polynomial in two variables—turns out to depend crucially on the ''genus'' of the curve. The ''genus'' can be defined as follows:<ref group="note">If we want to study the curve <math>y^2 = x^3 + 7</math>. We allow ''x'' and ''y'' to be complex numbers: <math>(a + b i)^2 = (c + d i)^3 + 7</math>. This is, in effect, a set of two equations on four variables, since both the real
and the imaginary part on each side must match. As a result, we get a surface (two-dimensional) in four-dimensional space. After we choose a convenient hyperplane on which to project the surface (meaning that, say, we choose to ignore the coordinate ''a''), we can
plot the resulting projection, which is a surface in ordinary three-dimensional space. It
then becomes clear that the result is a [[torus]], loosely speaking, the surface of a doughnut (somewhat
stretched). A doughnut has one hole; hence the genus is 1.</ref> allow the variables in <math>f(x,y)=0</math> to be complex numbers; then <math>f(x,y)=0</math> defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables, that is, four dimensions). If we count the number of (doughnut) holes in the surface; we call this number the ''genus'' of <math>f(x,y)=0</math>. Other geometrical notions turn out to be just as crucial.
There is also the closely linked area of [[Diophantine approximations]]: given a number <math>x</math>, then finding how well can it be approximated by rationals. (We are looking for approximations that are good relative to the amount of space that it takes to write the rational: call <math>a/q</math> (with <math>\gcd(a,q)=1</math>) a good approximation to <math>x</math> if <math>|x-a/q|<\frac{1}{q^c}</math>, where <math>c</math> is large.) This question is of special interest if <math>x</math> is an algebraic number. If <math>x</math> cannot be well approximated, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of [[Glossary of arithmetic and diophantine geometry#H|height]]) turn out to be critical both in Diophantine geometry and in the study of Diophantine approximations. This question is also of special interest in [[transcendental number theory]]: if a number can be better approximated than any algebraic number, then it is a [[transcendental number]]. It is by this argument that [[Pi|{{pi}}]] and [[e (mathematical constant)|e]] have been shown to be transcendental.
Diophantine geometry should not be confused with the [[geometry of numbers]], which is a collection of graphical methods for answering certain questions in algebraic number theory. ''Arithmetic geometry'', however, is a contemporary term
for much the same domain as that covered by the term ''Diophantine geometry''. The term ''arithmetic geometry'' is arguably used
most often when one wishes to emphasise the connections to modern algebraic geometry (as in, for instance, [[Faltings's theorem]]) rather than to techniques in Diophantine approximations.-->
==Lihat pula==
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