Class field theory can be divided into two parts, local and global. In each part it is the study of all the abelian extensions of a certain base field. The underlying philosophy is to describe all abelian extensions in terms of objects residing within, or close to, the base field.

... the geometry over p-adic fields, and more generally over complete local rings, can provide us only with local data; and the main tasks of algebraic geometry have always been understood to be of a global nature. It is well known that there can be no global theory of algebraic varieties unless one makes them "complete", by adding to them suitable "points at infinity," embedding them, for example, in projective spaces. In the theory of curves, for instance, one would not otherwise obtain such basic facts as that the number of poles and zeros of a function are equal, of that the sum of residues of a differential is 0.

... most physicists would probably agree that the place of local fields is nowhere so secure as in the theory of photons and gravitons, whose properties seem indissolubly linked with the space-time concepts of gauge invariance (of the second kind) and/or Einstein's equivalence principle.

The key insight of the approach is to consider that social action takes place in arenas, what may be called fields, domains, sectors, or organized social spaces... Fields contain collective actors who try to produce a system of domination in that space. To do so requires the production of a local culture that defines local social relations between actors.

It has been generally believed that only the complex numbers could legitimately be used as the ground field in discussing quantum-mechanical operators. Over the complex field, Frobenius' theorem is of course not valid; the only division algebra over the complex field is formed by the complex numbers themselves. However, Frobenius' theorem is relevant precisely because the appropriate ground field for much of quantum mechanics is real rather than complex.

The real field of knowledge is not the given fact about things as they are, but the critical evaluation of them as a prelude to passing beyond their given form. Knowledge deals with appearances in order to get beyond them. …. The concept of reality has thus turned into the concept of possibility. The real is not yet ‘actual,’ but is at first only the possibility of an actual. P. 145

In the theory of electric fields it is assumed that points have meaning. ...Physicists using general relativity... cannot speak of a point, except by naming some features of the field lines that will uniquely distinguish that point. ...the network of relationships evolve with time... constantly changing.

Arithmetic is the science of the Evaluation of Functions, Algebra is the science of the Transformation of Functions.

Perhaps the least inadequate description of the general scope of modern Pure Mathematics—I will not call it a definition—would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations.

The family of mathematical problems discussed here has emerged in recent years as a result of efforts to put a small chapter of quantum field theory, the so-called external field problem, on a sound mathematical footing. The external field problems is special because the partial differential equations for the unknown field is linear, but the coefficients are allowed to vary in space and time and that gives rise to some surprises, which seem to be of general interest. There is a vast and in large part turgid mathematical physics literature on the subject. To make the general wisdom which has accumulated there more readily available to a mathematical audience I have, in the following, tried to place the problems in their physical context, and still to bring out the essential mathematical questions many of which remain to be answered.

The field is the sole governing agency of the particle

The study of the geometry of a Galois space S_r,q, i. e. of a projective r-dimensional space over a Galois field of order q = p^h. where p, h are positive integers and p is a prime (the characteristic of the field), has recently been pursued and developed along new lines ... In it, both algebraic-geometric and arithmetical methods have been applied, including the use of electronic calculating machines; moreover, some of the problems dealt with are deeply connected with information theory, especially with the construction of q-ary error-correcting codes. It is actually a chapter of arithmetical geometry, which reduces to the investigation of certain questions on congruences mod p in the particular case when h = 1.

Since an algebraic function w(z) is defined implicitly by an equation of the form f(z,w) = 0, where f is a polynomial, it is understandable that the study of such functions should be possible by algebraic methods. Such methods also have the advantage that the theory can be developed in the most general setting, viz. over an arbitrary field, and not only over the field of complex numbers (the classical case).

Those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression.