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Conjugate element (field theory)

In mathematics, in particular field theory, the conjugate elements of an algebraic element α over a eld extension L / K, are the roots of the minimal polynomial P, and K, α of α over K. the paired elements are also called Galois conjugates or con ...

Euclidean field

In mathematics, a Euclidean field is an ordered field K for which every nonnegative element is a square, i.e. X ≥ 0 in K implies that X = Y 2 for some y in K.

Field trace

In mathematics, the field trace is a particular function defined relative to a finite field extension L / K which is a K-linear map from L to K.

P-adically closed field

In mathematics, in p-adically closed field is a field that has a property of closedness, which is a close analogue for P -adic fields that the closure of the real field. They were introduced by James ax and Simon B. Kochen in 1965.

P-basis

In algebra, a p-basis is a generalization of the notion of a separating transcendence basis of the extension field of characteristic P, introduced Teichmuller.

Purely inseparable extension

In algebra, a purely inseparable extension of fields is an extension K ⊆ K of a field of characteristic p &GT 0 such that every element of K is a root of equations of the form x y = a, where Q a power of P and K. a Purely inseparable extensions a ...

Pythagorean field

In algebra, Pythagorean is a field in which every sum of two squares is a square: equivalent to its Pythagoras number equal to 1. Pythagorean extension of the field F {\the style property display the value of f} is the continuation obtained by ad ...

Quadratically closed field

Field of real numbers is not quadratically closed, as it contains no square root of -1. Region of constructive quadratic number of closed but not algebraically closed. The Union of finite fields E 5 2 n {\F_ the style property display the value o ...

Rupture field

In abstract algebra, the gap field of the polynomial P {\the style property display the value of P} over a given field to a {\the style property display value To the}, such that P ∈ To } is the extension of the field K {\the style property displa ...

Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits or decomposes into linear factors.

Stufe (algebra)

In field theory, in Stufe from the field F is the smallest number of squares that sum to -1. If -1 cannot be written as a sum of squares, s = ∞ {\the style property display the value infty\}. In this case, F is a formally real field. Albrecht Pfi ...

Tensor product of fields

In abstract algebra, field theory has no direct product: the direct product of two fields, considered as ring, is not the field itself. However, it is often necessary to "merge" two fields K and L, either in cases where K and L are given as subfi ...

Ascending chain condition on principal ideals

In abstract algebra, the ascending chain condition can be applied on the belly of the left main, right main, or main bilateral ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals is satisfied if the ...

Ideal quotient

In abstract algebra, if I and J are ideals of a commutative ring R, their ideal relation is a set I: J = { r ∈ R ∣ r J ⊆ I } {\displaystyle I:J=\{r\in R\mid rJ\subseteq I\}} Then I: J is an ideal in R. the ideal factor is considered as a factor b ...

Krulls theorem

In mathematics, specifically in ring theory, Krulls theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal. Theorem was proved in 1929 Krull, which uses transfinite induction. The theorem admits a simple p ...

Nil ideal

In mathematics, more specifically ring theory, is left, right or two-sided ideal of a ring is called a nil ideal if each of its elements is nilpotent. In the nilradical of a commutative ring is an example of the zero ideal, in fact, this is the i ...

Nilpotent ideal

In mathematics, more specifically ring theory, an ideal I in a ring R is called nilpotent if there exists a natural number k such that K = 0. K and K is meant the additive subgroup generated by the set of all products of k elements, therefore, I ...

Real radical

In algebra, the real radical ideal I in a polynomial ring with real coefficients is the largest ideal containing I with the same vanishing locus. He plays a similar role in algebraic geometry that the radical of an ideal plays in algebraic geomet ...

Berlekamp–van Lint–Seidel graph

In graph theory, in Berlekamp–van lint–Seidel graph is locally linear strongly regular graph with parameters. This means that it has 243 vertices, 22 edges to vertex exactly one common neighbor, every pair of adjacent vertices, and exactly two co ...

Games graph

In graph theory, games, graphics is the largest known locally linear strongly regular graph. Its parameters are strongly regular graphs. This means that it has 729 vertices and edges 40824. Each edge has a unique triangle and every non-adjacent p ...

Sobolev conjugate

Sobolev conjugate of P for 1 ≤ p &lt N {\the style property display value 1\leq p ∗ = p n-n − n > p {\the style property display the value of n^{*}={\frats {PN}{n-p}}> p} This is an important parameter in the Sobolev inequalities.

Soucek space

In mathematics, Soucek spaces, generalizations of Sobolev spaces, the name of the Czech mathematician Jiei Soucek. One of their main advantages is that they offer a way of dealing with the fact that the Sobolev space W 1.1 is a reflexive space, a ...

Trace operator

In mathematics, the concept of trace operator plays an important role in the study of existence and uniqueness of solutions to boundary value problems, i.e. differential equations with given boundary conditions. The trace statement allows to exte ...

Four-momentum

In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional space-time. Momentum is a vector in three-dimensional space, similarly four-momentum is a vector in space-time. In the cont ...

Four-velocity

In physics, particularly special relativity and General relativity, the four velocity is a vector in four-dimensional space-time, which represents the relativistic analogue of velocity, which is a three dimensional vector in space. Physical event ...

Heaviside cover-up method

In Heaviside cover-up method, named after Oliver Heaviside, is one possible approach in determining the coefficients when performing the partial fraction decomposition of rational functions.

Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R, where U is an open subset of R n, which satisfies Laplaces equation, that is ∂ 2 f ∂ x 1 2 ...

Analytic subgroup theorem

In mathematics, the analytic subgroup theorem is a significant result in the modern theory of transcendental numbers. It can be viewed as a generalization of bakers theorem on linear forms of logarithms. Gisbert Wustholz proved it in the 1980-ies ...

Hypertranscendental number

A complex number is called hypertranscendental if its not the value at an algebraic point function which is the solution of an algebraic differential equation with coefficients in Z and with algebraic initial conditions. The term was coined by D. ...

Fine topology (potential theory)

In mathematics, in the field of potential theory, fine topology is a natural topology for setting the study of subharmonic functions. In the earliest studies of subharmonic functions, namely those for which Δ and U ≥ 0, {\the style property displ ...

Polar set (potential theory)

In mathematics, in the field of classical potential theory, polar sets are the "negligible sets", similar to that in which sets of measure zero are the negligible sets in measure theory.

Beltramis theorem

In mathematics - specifically, in Riemannian geometry - theorem Beltramis is a result named after the Italian mathematician Eugenio Beltrami which States that geodesic maps preserve the property of having constant curvature. More precisely, if tw ...

Myers–Steenrod theorem

The two theorems in the mathematical field of Riemannian geometry to bear the name of Myers–Steenrod theorem, as paper of 1939 Myers and Steenrod. The first States that any distance-preserving map between two connected Riemannian manifolds are ac ...

Nash embedding theorem

In our embedding theorems, named after John Forbes Nash, claim that every Riemannian manifold can be isometrically invested in Euclidean space. Isometric means preserving the length of each path. For instance, bending without stretching or tearin ...

Splitting theorem

The splitting theorem is a classical theorem in Riemannian geometry. It States that if a complete Riemannian manifold m with Ricci curvature R i c M ≥ 0 {\displaystyle {\rm {Ric}}M\geq 0} there is a straight line, i.e. a geodesic γ such that d γ ...

Roulette (curve)

Formally speaking, the curves must be differentiable curves in the Euclidean plane. A fixed curve is kept invariant curve rolling is subjected to continuous congruence transformation such that at all times the curves are tangent at the contact po ...

Astroid

In astroid is a particular mathematical curve: a hypocycloid with four doors. In particular, the locus of a point on a circle as it rolls inside a fixed circle with a radius. Double generation, it is also a place for a point on a circle as it rol ...

Centered trochoid

In geometry, a centered trochoid is the roulette formed by a circle rolling on another circle. That is, the path traced by a point attached to a circle, the circle rolls without slipping on a fixed circle. This term covers both epitrochoid and hy ...

Cyclogon

In mathematics, in geometry, cyclogon curve described by the vertices of the polygon that rolls without slipping along a straight line. There are no restrictions on the nature of the polygon. It can be a regular polygon as an equilateral triangle ...

Cycloid

The cycloid is the curve described by a point on the rim of the circumference of the wheel as the wheel rolls in a straight line without slipping. Cycloid is a specific form of a trochoid and an example of a roulette, a curve generated by a curve ...

Epitrochoid

In the epitrochoid the roulette traced is attached to the circumference of radius R rolls on the outside of a fixed circle of radius R, where the point is distance D from the center to the outer circumference. Parametric equations for the epitroc ...

Encyclopedic dictionary

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Pino - logical board game which is based on tactics and strategy. In general this is a remix of chess, checkers and corners. The game develops imagination, concentration, teaches how to solve tasks, plan their own actions and of course to think logically. It does not matter how much pieces you have, the main thing is how they are placement!

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