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[No title]   (Site not responding. Last check: 2007-10-13) In particular, if K is a field, R is a subring of K, gen_R K is finite, and R is a quotient of a Jacobson ring, then R is a field and len_R K is finite. Define F0 as the field of fractions of R0; F1 as the field of fractions of R1; F2 as the field of fractions of R2; and F3 as the field of fractions of R3. Thus F0[x1] is a field, and len_F F0[x1] is finite. cr.yp.to /zgk.html   (643 words)

Function Fields and their Elements Coerce the element g into the function field F of a scheme where g is some function on the scheme of F, for example, g may be an element of the field of fractions of the coordinate ring of a scheme having F as its function field. Given an element f of a function field of a scheme, return f as an element of the field of fractions of the coordinate ring of the scheme f is a function on. Given an element f of a function field of a (projective) scheme X, returns an element of the field of fractions of the coordinate ring of the ambient of X whose restriction to X as a rational function is f. magma.maths.usyd.edu.au /magma/htmlhelp/text1200.htm   (609 words)

PlanetMath: place of field is a valuation domain; so any place of a field determines a unique valuation domain in the field. Cross-references: kernel, ideal, unit, inverse, residue-class ring, onto, ring, homomorphism, canonical, maximal ideal, field of fractions, valuation domain, easy to see, ring homomorphism, restriction, subring, preimage, satisfies, mapping, field This is version 13 of place of field, born on 2005-01-13, modified 2006-12-12. planetmath.org /encyclopedia/SpotOfField.html   (157 words)

Field (mathematics) Summary A "field" is the name given to a pair of numbers and a set of operations which together satisfy several specific laws. For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F. Differential field, a field equipped with a derivation. www.bookrags.com /Field_(mathematics)   (2054 words)

Algebraic Function Fields [HB 53] A type for field of fractions of orders of algebraic function fields has been introduced. of an order of a function field now returns a sequence of elements in the field of fractions of the order rather than the function field. Fields of Fractions for orders of function fields have been introduced. www.math.lsu.edu /magma/rel/node26.htm   (168 words)

Springer Online Reference Works Basic problems in the theory of fields consist of giving a description of all subfields of a given field, of all fields containing a given field, i.e. Field theory originated (within the framework of the theory of algebraic equations) in the middle of the 19th century. The German term for  "field"  is  "Körper"  and this is of course the term used in [a2]. eom.springer.de /f/f040090.htm   (480 words)

KSORT$ 3R2A -- Reentrant Internal Sort A field consists of a whole or partial word of a record and may be sorted numerically or alphanumerically, in ascending or descending order. Long alphanumeric fields would thus be represented as several consecutive whole-word fields with partial-word fields at the beginning or end if necessary. Starting with the second field of the proc call, there must be a 3-part field describing each sort field (0 fractions and codes may be omitted). cgibin.rcn.com /leistlc/ksort$.htm   (2145 words)

Ideals and Quotients Given a (fractional) ideal I of O, return two elements of the field of fractions of O that generate I as an ideal. Given an integral ideal I of O, return two elements of the field of fractions of O that form a two-element normal presentation for I, as well as an integer g such that I is g-normal. The denominator of the fractional ideal I. This is the smallest positive integer d such that dI is an integral ideal. www.math.ufl.edu /help/magma/text391.html   (882 words)

Visual Fractions Information Page FRACTIONITIS frak'sheni'tis An increase in anxiety caused by the appearance of a numeral in fraction form. Cure: Associate fractions with a pleasant object, such as 1/4 of a pizza. Visual Fractions programs are being re-written to run with the FLASH plug in. www.visualfractions.com /inform.htm   (697 words)

4.3.4 Magnetic-field structure For this reason, the field ratios are essentially unconstrained there (Table 7). The component ratios for the spine are not well determined (Table 7) and could quite plausibly be identical to those for the shear layer. The arrows mark the location at the edge of the jet where the three field components are roughly equal, as discussed in the text. www.cv.nrao.edu /~abridle/3c31free/l2h/4_3_4_Magnetic_field_struct.html   (270 words)

WeatherWise Help The "help" menu in the upper left corner is able to give you a conversion chart for fraction to decimal conversions, a symbol chart for the types of doors you'll be entering, and an "about" box that tells you who made the program. For fractions of a foot, say for example the wall measures 40 1/2 feet wide, then enter the decimal equivalent for this number (40.5 in this case). You will notice that the "Wall Number" field cannot be edited during this process - this is to keep the same wall from being duplicated in the database under different numbers. www.wou.edu /~roengli/HelpFiles/wwWalls.html   (1221 words)

Continued Fractions - History Those who wish to study a particular field of mathematics, whether it be statistics, abstract algebra, or continued fractions, will first need to study their field's past. The origin of continued fractions is traditionally placed at the time of the creation of Euclid's Algorithm. This brief sketch into the past of continued fractions is intended to provide an overview of the development of this field. archives.math.utk.edu /articles/atuyl/confrac/history.html   (933 words)

Creation Functions Create the field F of rational functions in 1 indeterminate (consisting of quotients of univariate polynomials) over the integral domain R. The angle bracket notation may be used to assign names to the indeterminates, just as in the case of polynomial rings, e.g.: Explicit coercion is always allowed between function fields having the same number of variables (and suitable base rings), whether they are global or not, and the coercion maps the i-variable of one function field to the i-th variable of the other function field. Given the rational function field F (which is the field of fractions of the polynomial ring R), and polynomials a, b in R (with b != 0), construct the rational function a / b. www.umich.edu /~gpcc/scs/magma/text603.htm   (514 words)

Creation Functions A tower of fields similar to that of NumberField is created and the same restrictions as for that function apply to the polynomials that can be used in the constructor. Return the field containing all fractions of elements of O. The angle bracket notation can be used to assign names to the basis elements of F and assign these elements to variables, e.g. To specify homomorphisms from algebraic fields or orders in algebraic fields, it is necessary to specify the image of the generating elements, and possible to specify a map on the ground field. www.math.niu.edu /help/math/magmahelp/text663.html   (3737 words)

Springer Online Reference Works is a finite field, or else is a finite extension of is relatively prime with the characteristic of the field The theory of modules over a discretely-normed ring is very similar to the theory of Abelian groups [3]. eom.springer.de /d/d033180.htm   (328 words)

field - Search Results - MSN Encarta Field (mathematics), set of elements that can be operated on according to rules that satisfy certain properties. baseball, cricket, field hockey, Football, football, American, football, Gaelic, hurling, lacrosse, Little League, pictures of sports fields, polo,... meadow, pasture, grassland, lea, grazing, arena, playing field, turf, ground, sports ground, pitch, park, subject, area, topic, discipline, theme,... encarta.msn.com /encnet/refpages/search.aspx?q=field   (174 words)

WeatherWise Help For instance: if a wall is designated as being in area #1, then enter "1" in the field for Area Number when entering the measurements for the door that is a part of this wall. For fractions of an inch, say for example the door measures 40 1/2 inches wide, then enter the decimal equivalent for this number (40.5 in this case). As with the width, you must use the decimal equivalent for fractions of an inch rather than entering a fraction directly. www.wou.edu /~roengli/HelpFiles/wwDoors.html   (951 words)

[astro-ph/0608569] The DEEP2 Galaxy Redshift Survey: The evolution of the blue fraction in groups and the field At fixed z, there is also a correlation between blue fraction and galaxy magnitude, such that brighter galaxies are more likely to be red, both in groups and in the field. In terms of evolution, the blue fraction in groups and the field remains roughly constant from z=0.75 to z ~ 1, but beyond this redshift the blue fraction in groups rises rapidly with z, and the group and field blue fractions become indistinguishable at z ~ 1.3. The convergence between the group and field blue fractions at z ~ 1.3 implies that DEEP2 galaxy groups only became efficient at quenching star formation at z ~ 2; this result is broadly consistent with other recent observations and with current models of galaxy evolution and hierarchical structure growth. arxiv.org /abs/astro-ph/0608569   (369 words)

Creation Functions An important subring of a number field K is its ring of integers (O)_K, consisting of integral elements of the field, that is, elements that are roots of monic integer polynomials. Because it is sometimes convenient to view field elements with respect to some integral basis that is not the power basis (corresponding to the polynomial representation) we also allow the coercion of field elements into orders: this will return an order element together with an integer denominator, representing the field element. Coerce a into the field K. Here a may be an integer or a rational field element, or an element from a subfield of K, or from an order in such. www.math.uiuc.edu /Software/magma/text355.html   (2057 words)

Introduction The number field module in Magma is based on the current Kash release (Kant-V4) [KAN97], [KAN00], developed by the group of M. Pohst in Berlin. is a finite field extension of a number field k or Q constructed as a quotient ring of the univariate polynomial ring over the base field modulo some irreducible polynomial: K := k[t]/(f(t)k[t]) Or, the field may be constructed as a multivariate quotient: K := k[s_1,..., s_n]/(f_1(s_1),..., f_n(s_n)) where all the polynomials are univariate. If a field may be either a number field or field of fractions it will be referred to as an algebraic field. www.umich.edu /~gpcc/scs/magma/text630.htm   (840 words)

Element Operations   (Site not responding. Last check: 2007-10-13) Given an element a from a number field or order L, return the characteristic polynomial of the element over the subfield or suborder K; here K must be the field or order over which L is defined as an extension. For field elements the polynomial will have coefficients in the rational field, for order elements the coefficients will be in the ring of integers. Given an element a from a number field or order L, return the minimal polynomial of the element over the subfield or suborder K; here K must be the field or order over which L is defined as an extension. math.wisc.edu /help/magma/text448.html   (1601 words)

Newton polygon Information In the original case, the local field of interest was the field of formal Laurent series in the indeterminate X, i.e. After the introduction of the p-adic numbers, it was shown that the Newton polygon is just as useful in questions of ramification for local fields, and hence in algebraic number theory. A priori, given a polynomial over a field, the behaviour of the roots (assuming it has roots) will be unknown. www.bookrags.com /wiki/Newton_polygon   (453 words)

Abstract of Bernard Teissier's paper "Valuations, deformations ..."   (Site not responding. Last check: 2007-10-13) Given a noetherian local integral domain $R$ and a valuation $\nu$ of its field of fractions which is non negative on $R$, I study a geometric specialization of $R$ to the graded ring ${\rm gr}_\nu R$ determined by the valuation. I show that if the residue field of $R$ is algebraically closed, for 0-dimensional valuations this graded ring corresponds to an essentially toric variety, possibly of infinite embedding dimension, and consider some possible applications of this fact to local uniformization by deformation of a resolution of singularities of the toric variety. This leads to the study of the $\nu$-adic completion of a noetherian local domain and of the structure and regularity of certain semigroup algebras and graded algebras associated to valuations. math.usask.ca /fvk/Absteis1.html   (151 words)

EPA: Federal Register: Clopyralid; Pesticide Tolerance DowElanco, on September 27, l996, requested that the time-limited tolerances for residues of the herbicide clopyralid in the field corn commodities under the regulations mentioned above be made permanent tolerances based on residue data that they had submitted as required to change the tolerances from time-limited to permanent tolerances. These estimates are based on the assumption that 100% of the field corn commodities are derived from field corn cultured with the aid of the herbicide clopyralid. The quantitative limit of the method is 0.05 micrograms/gram in field corn fodder and forage and grain. www.epa.gov /fedrgstr/EPA-PEST/1997/April/Day-16/p9372.htm   (3639 words)

Exercises Prove that any finite subgroup of the multiplicative group of a field is cyclic. A is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. A field with the topology induced by a valuation is a topological field, i.e., the operations sum, product, and reciprocal are continuous. modular.fas.harvard.edu /129/ant/html/node90.html   (1075 words)

Ideals and Quotients Where an element or elements are returned from a function the elements are usually in the field of fractions if the ideal has the fractional type and in the order if it has integral type. Make the ideal I either fractional (first case where F is a field of fractions compatible with the order of I) or integral (second case where O is an order compatible with the order of I). The denominator of the fractional ideal I. This is the smallest positive integer d such that d * I is an integral ideal. www.math.niu.edu /help/math/magmahelp/text674.html   (2645 words)

C_F The origin of continued fractions is very difficult to pinpoint as we can find examples of these fractions throughout mathematics in the last 2000 years, but its true foundations were not laid until the late 1600's, early 1700's. The Indian mathematician Aryabhata used a continued fraction to solve a linear indeterminate equation. Infinite, where the terms just go on for ever, and finite continued fraction, where there are only a fixed number of terms. acm.uva.es /p/v105/10521.html   (399 words)

First year curriculum Definition of a field, field of fractions of an integral domain. Field extensions and Galois theory: separable and inseparable extensions, norm and trace, algebraic and transcendental extensions, transcendence basis, algebraic closure, fundamental theorem of Galois theory, solvability of equations, cyclotomic extensions and explicit computations of Galois groups. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. www.math.upenn.edu /grad/1stYearGrad.html   (895 words)

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