Italian school of algebraic geometry (4799 bytes)
1: ...etry still bear the names of those of the Italian school.
3: ...in all its refinements (via the detailed geometry of the [[theta-divisor]]).
5: ...vich]] school and others by around 1960. The form of the Riemann-Roch theorem on a surface was also wo...
7: ...n embryo form, and in some cases the articulation of those exceeded the available technical language.
9: ...pursuit of [[projective geometry]] as [[synthetic geometry]], which during the period under discussion was a...

Algebraic geometry and analytic geometry (3533 bytes)
1: ...lso in a sense included in the scope of algebraic geometry, that is not the topic being discussed in this ar...
3: ...rest is typically in going the other way. Results of this kind were proved first in the [[nineteenth c...
7: ...uch coverings all come from [[finite extension]]s of the [[function field]].
9: ...l in fact relate to a finitely-generated subfield of ''K''.
11: ...c methods within the classical parts of algebraic geometry.

Algebraic geometry (15675 bytes)
1: ...le of mathematics, both conceptually and in terms of technique.
3: == Zeroes of simultaneous polynomials ==
5: ...th>\mathbb R^3</math> could be defined as the set of all points <math>(x,y,z)</math> with
9: ...math>\mathbb R^3</math> can be defined as the set of all points <math>(x,y,z)</math> which satisfy th...
16: ...^n_k</math> is, for the moment, just a collection of points.

Motive (algebraic geometry) (5923 bytes)
1: In [[algebraic geometry]] the idea of a '''motive''' intuitively refers to
2: ...ms, as can currently be shown. While the category of motives was supposed to be the [[universal Weil c...
4: ...pted among mathematicians) point to the existence of general underlying framework. Some mathematician...
9: ...sponding motive [''X''], so the simplest examples of motives are:
18: * for the number of points over any [[finite field]]

List of algebraic geometry topics (4388 bytes)
1: This is a '''list of [[algebraic geometry]] topics''', by Wikipedia page.
3: ==Classical topics in [[projective geometry]]==
15: *[[Fundamental theorem of projective geometry]]
16: *[[Duality (projective geometry)]]
20: ==[[Algebraic curve]]s==

Algebraic K-theory (2872 bytes)
3: ...s]], '''algebraic K-theory''' is an advanced part of [[homological algebra]] concerned with defining a...
7: ...uite hard to compute (even when ''R'' is the ring of [[integer]]s).
9: ...tructed as a quotient of the multiplicative group of the field tensored with itself, with some explici...
11: ...ing a deep and difficult theory), by a definition of [[Daniel Quillen]]. Quillen defined
15: ..., ''B'' is the [[classifying space]] construction of [[homotopy theory]], and the ⁺ is Quil...

Algebraic enumeration (303 bytes)
3: ... the objects being counted can be taken advantage of to simplify the counting process. See also [[com...

Algebraic element (2538 bytes)
1: ...lynomial]]s are in [[abstract algebra]] called '''algebraic elements'''. They can be created in a larger stru...
3: ...t ''g''(''a'')=0. Elements of ''L'' which are not algebraic over ''K'' are called '''transcendental''' over '...
5: ...f [[complex number]]s and '''Q''' being the field of [[rational number]]s).
8: ...] is algebraic over '''Q''', since it is the root of the polynomial ''g''(''x'') = ''x''² -...
9: ...ndental over '''Q''' but algebraic over the field of [[real number]]s '''R'''.

Algebraic torus (1180 bytes)
1: ...ticular kind of [[algebraic group]], that becomes of the simple form
5: ...]]. They were so named by analogy with the theory of ''tori'' in [[Lie group]] theory (see [[maximal t...
7: ...field]]s, and should be qualified to take account of inseparability questions.)
13: [[Category:Algebraic groups]]

Algebraic surface (2696 bytes)
1: ...ed, however, in the [[Italian school of algebraic geometry]], and are up to 100 years old.
3: Examples of algebraic surfaces include:
12: * [[surfaces of general type]].
14: ... under which a point is replaced by the ''curve'' of all limiting tangent directions coming into it (a...
16: ...algebraic surface]]s. The ''general type'' class, of [[Kodaira dimension]] 2, is very large (degree 5 ...

Algebraic syntax (1736 bytes)
1: ... to the left and right and up arrows to the right of symbols.
3: ... Thus [A, B] [C, D] = [AC, BD]. In our groupoid of directed type strings we define X&rarr;&larr;X = ...
5: ...en a correct lexicon to begin with, the theory of algebraic syntax claims that the grammatical sentences will...
7: [Need to add explanation of how up and down arrows are used, as well as recur...

Algebraic curve (1950 bytes)
1: ...ebraic variety|dimension]] equal to 1. The theory of these [[curve]]s in general was quite fully devel...
3: ...nts include crossings over itself, and also types of ''cusp'', for example that shown by the curve wit...
5: ...ce]] (i.e. ''complete'' in the sense of algebraic geometry). If for example we simply look at a curve in the...
7: ...s theorem correct. This can serve as a definition of the genus for curves over other fields.
9: ... the [[Jacobian variety]], an [[abelian variety]] of dimension ''g''

Algebraic group (2535 bytes)
1: ...' is a [[group (mathematics)|group]] that is an [[algebraic variety]], such that the multiplication and inver...
3: Several important classes of groups are algebraic groups, including:
5: * GL(''n'','''C'''), the [[general linear group]] of invertible matrices over '''C'''
8: ...y (mathematics)|category]] of [[algebraic variety|algebraic varieties]].
10: ...oduct]] of an ''abelian variety'' with a ''linear algebraic group''.

Algebraic integer (1467 bytes)
1: ...'' is a [[complex number]] &alpha; that is a root of an equation
5: ...]] (that is, the coefficient of the largest power of ''x'' in ''P''(''x'') is one) with [[integer]] co...
7: ...t can be shown that not all algebraic numbers are algebraic integers.
9: ...atest common divisor]] of the set of coefficients of ''P'' is 1; note that this is weaker than requiri...
11: ... that is closed under the operation of extraction of roots.

Algebraic structure (4859 bytes)
1: ...re are only relations and no operations, we speak of a '''relational structure'''.
3: ...ding on the operations, relations and axioms, the algebraic structures get their names.
4: The following is a partial list of algebraic structures:
6: ...t, a set can itself be thought of as a degenerate algebraic structure, one that has zero operations defined o...
23: ...tion]] of scalar multiplication for every element of <i>R</i>, with an associativity condition linking...

Algebraic function (989 bytes)
1: In [[mathematics]], an '''algebraic function''' of [[indeterminates]]
9: ...'F'' is an [[implicit function]] that solves an [[algebraic equation]]. A simple example would be
13: ...her than symbolic expressions; however the term ''algebraic function'' is used nonetheless).

Algebraic number (5312 bytes)
2: ...'' is a solution of a [[polynomial]] [[equation]] of the form
4: ...x numbers which is most often called the field of algebraic numbers.
7: ...are also algebraic because they are the solutions of ''x''²&nbsp;&minus;&nbsp;2&nbsp;=&nbsp...
8: ...But most real numbers are not algebraic; examples of this are [[&pi;]] and ''[[e (mathematical constan...
9: ...for instance ''i'', the [[imaginary unit]], is an algebraic number since it satisfies

Algebraic extension (2866 bytes)
2: ...element of ''L'' is a [[root (mathematics)|root]] of some non-zero [[polynomial]] with coefficients in...
3: Field extensions which are not algebraic, i.e. which contain [[transcendental element]]s, ...
5: ...'C''']]/'''R''' and '''Q'''(&radic;2)/'''Q''' are algebraic.
7: ...is in turn implies that all finite extensions are algebraic.
9: ...raic number]]s is an infinite algebraic extension of the rational numbers.

Algebraic closure (2287 bytes)
2: ...y closed field|algebraically closed]]. It is one of many [[closure (mathematics)|closures]] in mathem...
4: ...re of ''K'', rather than ''an'' algebraic closure of ''K''.
6: ... be thought of as the largest algebraic extension of ''K''.
7: ...o ''L'' is contained within the algebraic closure of ''K''.
8: ...e algebraic closure of ''K'' is also the smallest algebraically closed field containing ''K'',

Algebraic variety (4129 bytes)
1: ...es. For varieties of algebras, and an explanation of the difference, see [[variety (universal algebra)...
3: ...ions allow you to do a lot of classical algebraic geometry.
5: ...cally ringed space, is isomorphic to a [[spectrum of a ring]].
6: ...initely generated ''K''-algebras, i.e., quotients of polynomial algebras modulo prime ideals.
9: ... usually not considered varieties, and we get rid of them by requiring the schemes underlying a variet...

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