Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.

He has made outstanding contributions to Langlands ' program in the fields of number theory and analysis, and in particular proved the Langlands conjectures for the automorphism group of a function field.

A semilinear transformation is a transformation which is linear " up to a twist ", meaning " up to a field automorphism under scalar multiplication ".

It is bijective and compatible with the arithmetical operations, and hence is a field automorphism.

From the point of view of Lie theory, the classical unitary group is a real form of the Steinberg group, which is an algebraic group that arises from the combination of the diagram automorphism of the general linear group ( reversing the Dynkin diagram A < sub > n </ sub >, which corresponds to transpose inverse ) and the field automorphism of the extension C / R ( namely complex conjugation ).

Over the finite field with q = p < sup > r </ sup > elements, F < sub > q </ sub >, there is a unique quadratic extension field, F < sub > q² </ sub >, with order 2 automorphism ( the rth power of the Frobenius automorphism ).

The eigenvalues of the automorphism of the torus generate an order of a real quadratic field, and the sol manifolds could in principle be classified in terms of the units and ideal classes of this order, though the details do not seem to be written down anywhere.

His idea was that if F is the Frobenius automorphism over the finite field, then the number of points of the variety X over the field of order q < sup > m </ sup > is the number of fixed points of F < sup > m </ sup > ( acting on all points of the variety X defined over the algebraic closure ).

A representation of a Lie group G on a vector space V ( over a field K ) is a smooth ( i. e. respecting the differential structure ) group homomorphism G → Aut ( V ) from G to the automorphism group of V. If a basis for the vector space V is chosen, the representation can be expressed as a homomorphism into GL ( n, K ).

Over an algebraically closed field, this and its triple cover are the only forms ; however, over other fields, there are often many other forms, or “ twists ” of E < sub > 6 </ sub >, which are classified in the general framework of Galois cohomology ( over a perfect field k ) by the set H < sup > 1 </ sup >( k, Aut ( E < sub > 6 </ sub >)) which, because the Dynkin diagram of E < sub > 6 </ sub > ( see below ) has automorphism group Z / 2Z, maps to H < sup > 1 </ sup >( k, Z / 2Z ) = Hom ( Gal ( k ), Z / 2Z ) with kernel H < sup > 1 </ sup >( k, E < sub > 6, ad </ sub >).

The vertex algebra of the conformal field theory describing bosonic string theory, compactified on the 24-dimensional quotient torus R < sup > 24 </ sup >/ Λ < sub > 24 </ sub > and orbifolded by a two-element reflection group, provides an explicit construction of the Griess algebra that has the monster group as its automorphism group.

) More generally, if F is any field of ( non-zero ) prime characteristic for which the Frobenius endomorphism is not an automorphism, F possesses an inseparable algebraic extension.

The isomorphism allows one to see the exotic outer automorphism of in terms of field automorphism and matrix operations.

In purely algebraic terms, for a given field K, this is the automorphism group over K of the field K ( X, Y ) of rational functions in two variables.

The full collineation group ( or automorphism group, or symmetry group ) is the projective linear group PGL ( 3, 2 ) which in this case is isomorphic to the projective special linear group PSL ( 2, 7 ) = PSL ( 3, 2 ), and the general linear group GL ( 3, 2 ) ( which is equal to PGL ( 3, 2 ) because the field has only one nonzero element ).

The field, then, is ripe for new Southerners to step to the fore and write of this twentieth-century phenomenon, the Southern Yankeefication: the new urban economy, the city-dweller, the pains of transition, the labor problems ; ;

Both the extent to which this is true and the limits of the field of perceptual skill involved should be acknowledged.

`` It would be a disgrace, and, as I have already said to the people of Tennessee, if Hearst is nominated, we may as well pen a dispatch, and send it back from the field of battle: ' All is lost, including our honor ' ''.

Some historians have found his point of view not to their taste, others have complained that he makes the Tory tradition appear `` contemptible rather than intelligible '', while a sympathetic critic has remarked that the `` intricate interplay of social dynamics and political activity of which, at times, politicians are the ignorant marionettes is not a field for the exercise of his talents ''.

One's daily work becomes sacred, since it is performed in the field of influence of the moral law, dealing as it does with people as well as with matter and energy.

To obey the moral law is just ordinary common sense, applied to a neglected field.

Such is the field for exercising our reverence.

In the field of political values, it is certainly true that students are not radical, not rebels against their parents or their peers.

Although because of the important achievements of nineteenth century scholars in the field of textual criticism the advance is not so striking as it was in the case of archaeology and place-names, the editorial principles laid down by Stevenson in his great edition of Asser and in his Crawford Charters were a distinct improvement upon those of his predecessors and remain unimproved upon today.

For it is their catastrophic concept of the Anglo-Saxon invasions rather than Kemble's gradualist approach which dominates the field.

The national average is more than $4 and that figure is considered by experts in the mental health field to be too low.

This is one of the most constructive suggestions made in this critical field in years, and I certainly hope it sparks some action.

A small business is defined as one which is independently owned and operated and which is not dominant in its field.

The Office of Foreign and Domestic Commerce is also active in the field of international trade, assisting Rhode Island firms in developing and enlarging markets abroad.

We do not favor one field over another: we think that all inquiry, all scholarly and artistic creation, is good -- provided only that it contributes to a sense and understanding of the true ends of life, as all first-rate scholarship and artistic creation does.

Recently added is the Brown & Sharpe turret drilling machine which introduces the company to an entirely new field of tool development.

Under the auspices of the Women's Recreation Association, interclass competition is organized in badminton, basketball, field hockey, golf, tennis, and swimming.

Mossberg's latest contribution to the field is the Model 500 ( from $73.50 ) ; ;

It is usually helpful to make a sketch map in the field, showing the size and location of the features of interest and to take photographs at the site.

Your competition is now proportionately greater -- you are competing not only against manufacturers in the same field but also against a vast array of manufacturers of other appealing consumer products.

New to the field is a duplex type butyrate laminate in which the two sheets of the laminate are of different color.

If V is a vector space over the field F, the general linear group of V, written GL ( V ) or Aut ( V ), is the group of all automorphisms of V, i. e. the set of all bijective linear transformations V → V, together with functional composition as group operation.

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F are those of degree one.

The sum, difference, product and quotient of two algebraic numbers is again algebraic ( this fact can be demonstrated using the resultant ), and the algebraic numbers therefore form a field, sometimes denoted by A ( which may also denote the adele ring ) or < span style =" text-decoration: overline ;"> Q </ span >.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.

Magnetic field lines of a ring current of finite diameter.

Provided the underlying scalars form a field ( more generally, a commutative ring with unity ), the definition below shows that such a function exists, and it can be shown to be unique.

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.

For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.

The center of a division ring is commutative and therefore a field.

Especially, the fact that the integers and any polynomial ring in one variable over a field are Euclidean domains such that the Euclidean division is easily computable is of basic importance in computer algebra.

* K, the ring of polynomials over a field K. For each nonzero polynomial P, define f ( P ) to be the degree of P.