(Trdgheitskorper) for p: next after this comes another field of still lower order called the resolving field (Zerlegungskiirper) for p, and in this field there is a. prime of the first degree, pm, such that pi+1=p'°, where k=m/mi. In the field of inertia pq; remains a prime, but becomes of higher degree; in Sli 1, which is called the branch-field (Verzweigungskorper) it becomes a power of a prime, and by going on in this way from the resolving field to Sl, we obtain (H-2) representations for any prime ideal of the resolving field. By means of these theorems, Hilbert finds an expression for the exact power to which a rational prime p occurs in the discriminant of Q, and in other ways the structure of S2 becomes more evident. It may be observed that whem m is prime the whole series reduces to SZ and the rational field, and we conclude that every prime ideal in S2 is of the first or mth degree: this is the case, for instance, when m=2, and is one of the reasons why quadratic fields are comparatively so simple in character. 52. Quadratic Fields.-Let rn be an ordinary integer different from +I, and not divisible by any square: then if x, y assume all ordinary rational values the expressions x+y/m are the elements of a field which may be called Q(~/ rn). It should be observed that y/rn means one definite root of x2-m=o, it does not matter which: it is convenient, however, to agree that ylm is positive when m is positive, and ii/m is negative when rn is negative. The principal results xgating to SZ will now be stated, and will serve as illustrations of 44'5¥-

In the notation previously used

n=[I, %(I+#711)] or [I, y/rn]

according as mal (mod 4) or not. In the first case A=m, in the second A =4m. The field S2 is normal, and every ideal prime in it is of the first degree.

Let q be any odd prime factor of m; then q=q2, where q is the rime ideal [q, %(q+/m)] when mel (mod 4) and in other cases ii, /m]. An odd prime p of which rn is a quadratic residue is the roduct of two prime ideals p, p', which may be written in the form Fp, § (a+/m)], [p, %(a-/m)] or Ip, a+/ml, [p, a-»/m], according as mal (mod 4) or not: here a is a root of x2Em (mod p), taken so as to be odd in the first of the two cases. All other rational odd primes are primes in Q. For the exceptional prime 2 there are four cases to consider: (i.) if msl (mod 8), then 2 = l2, %(I +/m)]X[2, $(I -/m)]. (ii.) If mE5 (mod 8), then 2 is prime: (iii.) if mE2 (mod 4), 2 = [2, / m]2: (iv.) if rn =3 (mod 54), 2 =[2, I +x/m)2. Illustrations will be found in § 44 for the case rn =23.

53. Normal Residues. Genera.-Hilbert has introduced a very convenient definition, and a corresponding symbol, which is a generalization of Legendre's quadratic character. Let n, m be rational integers, rn notasquare, 'wany rational prime; we write = +I if, to the modulus w, n is congruent to the norm of an integer contained in SZ(/ m); in all other cases we put = -1. This new symbol obeys a set of laws, among which may be especially noted = = and = +I, whenever n, m are prime

to p.

Now let ql, gi, . . qi be the different rational prime factors of the discriminant of SDH m); then with any rational integer a we may associate the l symbols

(a, nz) (11, (zz rn)

T, T2, —gt

and call them the total character of a with respect to SZ. This definition may be extended so as to give a total character for every ideal a in Q, as follows. First let S2 be an imaginary field (rn <o); we put r =t, 1'i=N(a), and call

rf-I m> <”- “>

gl ' ° i ' gr

the total character of a. Secondly, let Sl be a real field; we first determine the t separate characters of - 1, and if they are all positive we put 7z= -l-N(a), r=t, and adopt the r characters just written above as those of a. Suppose, however, that one of the characters of - I is negative; without loss of generality we may take it to be that with reference to qt. We then put r=t-I, i= =f=N(a) taken with such a sign that = +I, and takeas theltotal character of a the symbols for i=I, 2, . .(t-1).

With these definitions it can be proved that all ideals of the same class have the same total character, and hence there is a distribution of classes into genera, each genus containing those classes for which the total character is the same (cf. § 36).

Moreover, we have the fundamental theorem that an assigned set of r units =t I corresponds to an actually existing genus if, and only if, their product is +I, so that the number of actually existing genera is 2"'. This is really equivalent to a theorem about quadratic forms first stated and proved by Gauss; the same may be said about the 8 »-

5 /

next proposition, which, in its natural order, is easily proved by the method of ideals, whereas Gauss had to employ the theory of ternary quadratics.

Every class of the principal genus is the square of a class. An ambiguous ideal in S2 is defined as one which is unaltered by the change of / rn to -w/ rn (that is, it is the same as its conj ugate) and not divisible by any rational integer except =*=I. The only ambiguous prime ideals in S2 are those which are factors of its discriminant. PuttingA=q12 of. . .q,2, there are in S2 exactly 2t ambiguous ideals: namely, those factors of A, including n, which are not divisible by any square. It is a fundamental theorem, first proved by Gauss, that the number of ambiguous classes is equal to the number of genera.

54. Class-Number.-The number of ideal classes in the field (ZH m) may be expressed in the following forms:-

(i.) m <0:

h=i2 (n=1, 2.., ., -A);

(ii.) m>o= "

II sin E

h ;1 A

"2 log e Og avr

II sin Z-

In the first of these formulae -r is the number of units contained in Sl; thus r=6 for A= -3, r=4 for A= -4, -r=2 in other cases. In the second formula, e is the fundamental unit, and the products are taken for all the numbers of the set (I, 2, . . .A) for which = +I, =-'I respectively. In the ideal theory the' only way in which these formulae have been obtainediis by a modification of Dirichlet's method; to prove them without the use of transcendental analysis would be a substantial advance in the theory.

55. Suppose that any ideal in S2 is expressed in the form [wh wg]-; then any element of it is expressible as xw1+yw2, where x, y are rational integers, and we shall have N (xw1+;vw2) =ax2+bxy+cy', where a, b, c are rational numbers contained in the ideal. If we put x=a.x'+By', y='yx'--6y', where a, 6, 'y, 5 are rational numbers such that o.5-/3'y= il, we shall have simultaneously (a, b, c) (x, y)2 = (a', b', c') (x', y')2 as in § 52 and also V (af, ff, 6') (x', y')2 = N{x'(v.w1+'yw2) 'i'y'(/3w1+5w2)} =N(x'w'i-I-y'w'2), where [w'1, w'2l is the same ideal as before. Thus all equivalent forms are associated with the same ideal, and the numbers representable by forms of a particular class are precisely those which are norms of numbers belonging to the associated ideal. Hencethe class-number for ideals in SZ is also the class-number for a set of quadratic forms; and it can be shown that all these forms have the same determinant A. Conversely, every class of forms of determinant A can be associated 'with a definite class of ideals in S'l(~/ rn), where m=A or +A as the case may be. Composition of form-classes exactly corresponds to the multiplication of ideals: hence the complete analogy between the two theories, so long as they are really in contact. There is a corresponding theory of forms in Connexion with a field of order rz: the forms are of the order n, but are only very special forms of that order, because they are algebraically resolvable into the product of linear factors.

56. Complex Quadratic Forms.-Dirichlet, Smith and others, have discussed forms (a, b, c) in which the coefficients are complex integers of the form m-l-ni; and Hermite has considered bilinear forms oxx' +bxy'+b'x'y+cyy', where x', y', b' are the conjugates of x, y, b and a, e, are real. Ultimately these theories are connected with fields of the fourth order; and of course in the same way we might consider forms (a, b, c) with integral coefficients belonging to any given field of order n: the theory would then be ultimately connected with a field of order 2n.

57. Kronecker's Method.—In practice it is found convenient to combine the method of Dedekind with that of Kroneckcr, the main principles of which are as follows. Let F(x, y, 2, . . .) be a polynomial in any number of indeterminate (umbrae, as Sylvester calls them) with ordinary integral coefficients; if n is the greatest common measure of the coefficients, we have F =r1E, where E is a primary or unit form. The positive integer n is called the divisor of F; and the divisor of the product of two forms is equal to the product of the divisors of the factors. Next suppose that the coefficients of F are integers in a field S2 of order n. Denoting the conjugate forms by F', F", . . . F<"">, the product FF'F” . . . F<"'U=fE, where f is a real positive integer, and E a unit form with real integral coefficients. The natural numberf is called the norm of F. If F, G are any two forms (in SZ) we have N(FG)=N(F)N(G). Let the coefficients of F be o.1, ag, &c., those of G 61, BQ, &c., and those of FG Yu 72, &c.; and let p be any prime ideal in Q. Then if p"' is the highest power of p Contained in each of the coefficients a¢, and p” the highest power of p contained in each of the coefficients Bi, p'"+" is the highest ower of p contained by the whole set of coefficients vi. Vi/riting dvfinu, a2, . . .) for the highest ideal divisor of ai, az, &c.,

this is called the content of F; and we have the theorem that the