Basal Notions

BASAL NOTIONS.

Domain Rationality.— One of the most fundamental notions in Galois' theory is that of a domain of rationality which was first clearly formulated by Abel. When an equation f(x) =--aan + + . . . + a.- – + (1) offers itself for solution, its coefficients are sup known. It often happens that other quantities are known, or are assumed as known. Suppose A. . w are such quantics. finite in number. The totality of rational functions of these letters with rational numbers as coefficients constitutes a domain of rationality which we denote by, • • • w).

Thus any element of this domain may be ob tained by a finite number of additions, sub tractions, multiplications and divisions per formed on the letters A, p, ... (J. The domain of rationality which we lay at the base of a given algebraic investigation is to some degree a matter of choice. In any case, however, the coefficients of the equations we start with should lie in it.

Every domain must contain the domain R(1), called the absolute domain, and which is simply the totality of rational numbers. For the domain R(X . . . u) must contain the ele ment A/X=1. It is often desirable to add cer tain elements . . . to a domain R(X, p,.. .) forming the new domain R'(t, i, . . . e, C, . . .) The elements up C, ... are said to be adjoined to R.

Rational Functions in In elementary algebra and in the function theory a rational function of x, y, z, • • . • is an expression of the form .4x-florae's.' . . .

(2 Brsly ) ozo . ' where the exponents m, n are non-negative integers and the coefficients are merely inde pendent of the variables x, y, z, . . . In Galois' theory the term rational function is a much narrower one. In fact the term rational has no meaning unless in connection with a specific domain of rationality. Thus the expression (2) is a rational function of x, y, . . . in Galois' theory with respect to the doAlain R, when and only when the coefficients A, B, . . . lie in R. Thus such a function as (2) may be rational with respect to one domain and not with respect to another. For example x + V — 3y.

is a rational function of x, y with respect to the 2,ri domain R (p), p-=e 3 ; but it is not rational with respect to R(1) or R(V-5). When the denominator in (2) reduces to a constant, it becomes a rational integral function.

An equation as (1) is rational with respect to R when its coefficients a., . an lie in R.

Reducibility and Irreducibility is another basal notion of Galois' theory. The rational

integral function of x, y, s . . . , F(x, y, s, ...)=Axntlynitro...1-...+Lrlylso..., (3) with respect to the domain R is reducible in R when it is the product of two or more rational integral functions of x, y . . . with coefficients in R, viz., F=G-1-1•1... In this case we say F is divisible by G, H, . . . , which are factors or divisors of F in the domain R. If the ex (3) cannot be split up into two such factors, it is irreducible with respect to R. An equation as (1) is reducible or irreducible in R according as its left side is reducible or irre ducible in R.

An equation as (1) may be irreducible in one domain and reducible in another. Thus sz + x + 1=0 is irreducible in R(1), but is reducible in R (V-3). In fact, t =(x—p) (x— p e3 If el, . 6, are the roots of (1), it is obviously reducible in R (et, .. In fact its left side splits up into rational lineal factors, f (x) = as (x— fl) ... (x — en).

A theorem of utmost importance in Galois' theory is the following: Let f(x)— 0, g(x)0 be rational equations for the domain R, and let f(x)=0 be irredu cible in R. If g(x)—Oadmits a root of 0, it admits all the roots of f(x)11, and g(x) is divisible by f(x).

Equality.—As a third pillar on which Galois' theory rests is the distinction between formal and numerical equality, as we may designate it for lack of better terms. It is only by such a distinction that Galois was able to extend Lagrange's methods so as to apply to any type of algebraic equation. As long as we are deal ing with constants, equality and inequality are of course the same as in arithmetic — they are numerical. What do we mean, however, by the equation 4,(p, q, ) =,p(p, q, .).

0, IP being rational functions of the variables p, q, . .. for a domain R? In general R will contain variable elements which then may enter the coefficients of 0, *. Let us write the above equation ... ci, Cs, . . . . ci, c,,. • .), where m, m represent now all the variable elements in. 0, '1', among which will be p, while ci, cs, . . . represent constants. By an equation of the above type we mean : that for each and every set of numerical values m, . . . can take on consistent with their definition, the resulting numerical value of 0 is identical with that of When no two of the quantities 0, x, . . .

are equal we shall call them distinct or unequal.