A field is an commutative ring (F,+,*) of which every nonzero element is invertible. Over a field, we can perform addition, subtraction, multiplication and division.
The abelian group of non-zero elements of a field F is typically denoted by F×;
The characteristic of the field F is the smallest positive integern such that n·1 = 0; here n·1 stands for n summands 1 + 1 + 1 + ... + 1. If no such n exists, we say the characteristic is zero. Every non-zero characteristic is a prime number. For example, the rational numbers, the real numbers and the p-adic numbers have characteristic 0, while the finite field Zp has characteristic p.
A subfield of a field F is a subset of F which is closed under the field operation + and * of F and which, with these operations, forms itself a field.
If an element α of an extension field E over F is the root of a polynomial in F[x], then α is algebraic over F. If every element of E is algebraic over F, then E is an algebraic extension of F.
A field homomorphism between two fields E and F is a function
f : E → F
such that
f(x + y) = f(x) + f(y)
and
f(xy) = f(x) f(y)
for all x, y in E, as well as f(1) = 1. These properties imply that f(0) = 0, f(x-1) = f(x)-1 for x in E with x ≠ 0, and that f is injective. Fields, together with these homomorphisms, form a category. Two fields E and F are called isomorphic if there exists a bijective homomorphism
f : E → F.
The two fields are then identical for all practical purposes; however, not necessarily in a unique way. See, for example, complex conjugation.
The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers. Their detailed properties are studied in algebraic number theory.
The automorphism group of a Galois extension. When it is a finite extension, this is a finite group of order equal to the degree of the extension. Galois groups for infinite extensions are profinite groups.
The subject in which symmetry groups of differential equations are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Lie founded the theory of Lie groups. It has not, probably, reached definitive form.
A theory of thunderstorm charge separation based upon the suggested occurrence of the Lenard effect in thunderclouds, that is, the separation of electric charge due to the breakup of water drops.
An avalanche cannot possibly begin until the local electric field strength is high enough to accelerate a free electron to the minimum ionizing speed in the space and time interval corresponding to one mean free path of the electron, for upon collision, the electron usually loss its forward motion in the direction of the field.
According to this theory, the lower negative charge of a thundercloud is generated by the accumulation there of raindrops which have captured predominantly negative ions in their descent through the cloud.
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