Suppose that E is a finite extension of a field F of characteristic 0. If D is a
ID: 1941874 • Letter: S
Question
Suppose that E is a finite extension of a field F of characteristic 0. If D
is an integral domain that is a subring of E, then for each:
belonging to Gal (E/F), (Galois group of E over F), the image (D) is an integral domain that is isomorphic to D (you do not need to prove this, it follows from the definition of being an automorphism).
Show that is irreducible in D if and only if () is irreducible in (D).
Explanation / Answer
In a finite field there is necessarily an integer n such that 1 + 1 + ··· + 1 (n repeated terms) equals 0. It can be shown that the smallest such n must be a prime number, called the characteristic of the field. If a (necessarily infinite) field has the property that 1 + 1 + ··· + 1 is never zero, for any number of summands, such as in Q, for example, the characteristic is said to be zero. A basic class of finite fields are the fields Fp with p elements (p a prime number): Fp = Z/pZ = {0, 1, ..., p - 1}, where the operations are defined by performing the operation in the set of integers Z, dividing by p and taking the remainder; see modular arithmetic. A field K of characteristic p necessarily contains Fp,[6] and therefore may be viewed as a vector space over Fp, of finite dimension if K is finite. Thus a finite field K has prime power order, i.e., K has q = pn elements (where n > 0 is the number of elements in a basis of K over Fp). By developing more field theory, in particular the notion of the splitting field of a polynomial f over a field K, which is the smallest field containing K and all roots of f, one can show that two finite fields with the same number of elements are isomorphic, i.e., there is a one-to-one mapping of one field onto the other that preserves multiplication and addition. Thus we may speak of the finite field with q elements, usually denoted by Fq or GF(q). A subfield is, informally, a small field contained in a bigger one. Formally, a subfield E of a field F is a subset containing 0 and 1, closed under the operations +, -, · and multiplicative inverses and with its own operations defined by restriction. For example, the real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers and the rational numbers are examples. The notion of field extension lies at the heart of field theory, and is crucial to many other algebraic domains. A field extension F / E is simply a field F and a subfield E ? F. Constructing such a field extension F / E can be done by "adding new elements" or adjoining elements to the field E. For example, given a field E, the set F = E(X) of rational functions, i.e., equivalence classes of expressions of the kind where p(X) and q(X) are polynomials with coefficients in E, and q is not the zero polynomial, forms a field. This is the simplest example of a transcendental extension of E. It also is an example of a domain (the ring of polynomials in this case) being embedded into its field of fractions . The ring of formal power series is also a domain, and again the (equivalence classes of) fractions of the form p(X)/ q(X) where p and q are elements of form the field of fractions for . This field is actually the ring of Laurent series over the field E, denoted . In the above two cases, the added symbol X and its powers did not interact with elements of E. It is possible however that the adjoined symbol may interact with E. This idea will be illustrated by adjoining an element to the field of real numbers R. As explained above, C is an extension of R. C can be obtained from R by adjoining the imaginary symbol i which satisfies i2 = -1. The result is that R[i]=C. This is different from adjoining the symbol X to R, because in that case, the powers of X are all distinct objects, but here, i2=-1 is actually an element of R.