We can do this through the characteristic values and vectors of T in certain special cases, i.e., when the minimal polynomial for T factors over the scalar field F into a product of distinct monic polynomials of degree 1.

The second situation is illustrated by the operator T on Af ( F any field ) represented in the standard basis by Af.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.

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.

Also, no finite field F is algebraically closed, because if a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > n </ sub > are the elements of F, then the polynomial ( x − a < sub > 1 </ sub >)( x − a < sub > 2 </ sub >) ··· ( x − a < sub > n </ sub >) + 1

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

Given a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:

The slope field of F ( x ) = ( x < sup > 3 </ sup >/ 3 )-( x < sup > 2 </ sup >/ 2 )- x + c, showing three of the infinitely many solutions that can be produced by varying the Constant of integration | arbitrary constant C.

Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function

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.

The curl of a vector field F, denoted by curl F or ∇ × F, at a point is defined in terms of its projection onto various lines through the point.

If φ is a scalar valued function and F is a vector field, then

where F is the underlying field of the vector space being considered.

The National Institute of Standards and Technology at Boulder, Colorado has chosen to consider the field of cryogenics as that involving temperatures below − 180 ° C (- 292 ° F or 93. 15 K ).

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.

this is really a deficiency in the scalar field, namely, that it is not algebraically closed.

From these results, one sees that the study of linear operators on vector spaces over an algebraically closed field is essentially reduced to the study of nilpotent operators.

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.

The same argument proves that no subfield of the real field is algebraically closed ; in particular, the field of rational numbers is not algebraically closed.

Another example of an algebraically closed field is the field of ( complex ) algebraic numbers.

This can be rephrased by saying that the field of algebraic numbers is algebraically closed.

In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.

A field with no proper algebraic extensions is called algebraically closed.

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

The algebraic closure of K is also the smallest algebraically closed field containing K,

because if M is any algebraically closed field containing K, then the elements of M which are algebraic over K form an algebraic closure of K.

* There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers ; these are the algebraic closures of transcendental extensions of the rational numbers, e. g. the algebraic closure of Q ( π ).

The central idea of Diophantine geometry is that of a rational point, namely a solution to a polynomial equation or system of simultaneous equations, which is a vector in a prescribed field K, when K is not algebraically closed.