Page:EB1911 - Volume 19.djvu/889

(Trägheitskörper) for 𝔭: next after this comes another field of still lower order called the resolving field (Zerlegungskörper) for 𝔭, and in this field there is a prime of the first degree, 𝔭_𝑙＋1, such that 𝔭_𝑙＋1＝𝔭^𝒌, where 𝒌＝𝑚 ∕𝑚_𝑙. In the field of inertia 𝔭_𝑙＋1 remains a prime, but becomes of higher degree; in Ω_𝑙—1, which is called the branch-field (Verzweigungskörper) it becomes a power of a prime, and by going on in this way from the resolving field to Ω, we obtain (𝑙＋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 Ω, and in other ways the structure of Ω becomes more evident. It may be observed that whem 𝑚 is prime the whole series reduces to Ω and the rational field, and we conclude that every prime ideal in Ω 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 𝑚 be an ordinary integer different from ＋1, 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 Ω(√𝑚). It should be observed that √𝑚 means one definite root of 𝑥²—𝑚＝0, it does not matter which: it is convenient, however, to agree that √𝑚 is positive when 𝑚 is positive, and 𝑖√𝑚 is negative when 𝑚 is negative. The principal results relating to Ω will now be stated, and will serve as illustrations of §§ 44-51.

In the notation previously used

𝔬＝[1, ⁠1/2⁠(1＋√𝑚)] or [1, √𝑚]

according as 𝑚≡1 (mod 4) or not. In the first case 𝚫＝𝑚, in the second 𝚫＝4𝑚. The field Ω 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 prime ideal [𝒒, ⁠1/2⁠(𝒒＋√𝑚)] when 𝑚≡1 (mod 4) and in other cases [𝒒, √𝑚]. An odd prime 𝒑 of which 𝑚 is a quadratic residue is the product of two prime ideals 𝔭, 𝔭′, which may be written in the form [𝒑, ⁠1/2⁠(𝒂＋√𝑚)], [𝒑, ⁠1/2⁠(𝒂—√𝑚)] or [𝒑, 𝒂＋√𝑚], [𝒑, 𝒂—√𝑚], according as 𝑚≡1 (mod 4) or not: here 𝒂 is a root of 𝑥²≡𝑚 (mod 𝒑), taken so as to be odd in the first of the two cases. All other rational odd primes are primes in Ω. For the exceptional prime 2 there are four cases to consider: (i.) if 𝑚≡1 (mod 8), then 2＝[2,⁠1/2⁠(1＋√𝑚)]×[2,⁠1/2⁠(1—√𝑚)]. (ii.) If 𝑚≡5 (mod 8), then 2 is prime: (iii.) if 𝑚≡2 (mod 4), 2＝[2,√𝑚]²: (iv.) if 𝑚＝3 (mod≡4), 2＝[2,1＋√m)². Illustrations will be found in § 44 for the case 𝑚＝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, 𝑚 not a square, 𝑤 any rational prime; we write $\left({n,\ m \over w}\right)=+1$ if, to the modulus 𝑤, 𝒏 is congruent to the norm of an integer contained in Ω(√𝑚); in all other cases we put $\left({n,\ m \over w}\right)=-1$. This new symbol obeys a set of laws, among which may be especially noted $\left({n,\ w \over w}\right)=\left({w,\ n \over w}\right)=\left({n \over w}\right)$ and $\left({n,\ m \over w}\right)=+1$, whenever 𝒏, 𝑚 are prime to 𝒑.

Now let 𝒒 ₁, 𝒒 ₂ , . . . 𝒒_𝑡 be the different rational prime factors of the discriminant of Ω(√𝑚); then with any rational integer 𝒂 we may associate the 𝑡 symbols

$\left({a,\ m \over q_{1}}\right),\ \ \left({a,\ m \over q_{2}}\right),\ .\ .\ .\ \left({a,m \over q_{t}}\right)$

and call them the total character of 𝒂 with respect to Ω. This definition may be extended so as to give a total character for every ideal 𝔞 in Ω, as follows. First let Ω be an imaginary field (𝑚＜0); we put 𝒓＝𝑡, 𝒏＝𝖭(𝔞), and call

$\left({{\overline {n}},\ m \over q_{1}}\right),\ .\ .\ .\ \left({{\overline {n}},\ m \over q_{r}}\right)$

the total character of 𝔞. Secondly, let Ω be a real field; we first determine the 𝑡 separate characters of —&#;8198;1, and if they are all positive we put \overline{𝒏}＝ ＋𝖭(𝔞), 𝒓＝𝑡, and adopt the 𝒓 characters just written above as those of 𝔞. Suppose, however, that one of the characters of —1 is negative; without loss of generality we may take it to be that with reference to 𝒒_𝑡. We then put 𝒓＝𝑡—1, 𝒏＝±𝖭(𝔞) taken with such a sign that $\left({{\overline {n}},\ m \over q_{t}}\right)=+1$, and take as the total character of 𝔞 the symbols $\left({{\overline {n}},\ m \over q_{i}}\right)$ for 𝑖＝1, 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 ±1 corresponds to an actually existing genus if, and only if, their product is ＋1, so that the number of actually existing genera is 2^𝒓—1. This is really equivalent to a theorem about quadratic forms first stated and proved by Gauss; the same may be said about the 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 Ω is defined as one which is unaltered by the change of √𝑚 to — √𝑚 (that is, it is the same as its conjugate) and not divisible by any rational integer except ±1. The only ambiguous prime ideals in Ω are those which are factors of its discriminant. Putting 𝚫＝𝔮₁² 𝔮₂² . . . 𝔮_𝑡², there are in Ω exactly 2^𝑡 ambiguous ideals: namely, those factors of 𝚫, including 𝔬, 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 Ω(√𝑚) may be expressed in the following forms:—

(i.) 𝑚＜0:

$h={\tau \over 2\Delta _{n}}\Sigma \left({\Delta \over n}\right)n\qquad (n=1,\,2,\,.\,.\,.,\,-\Delta )$

(ii.) 𝑚＞0:

$h={1 \over 2\log \epsilon }\log {\Pi \sin {b\pi \over \Delta } \over \Pi \sin {a\pi \over \Delta }}$

In the first of these formulae 𝜏 is the number of units contained in Ω; thus 𝜏＝6 for 𝚫＝—3, 𝜏＝4 for 𝚫＝—4, 𝜏＝2 in other cases. In the second formula, 𝜖 is the fundamental unit, and the products are taken for all the numbers of the set (1, 2, . . . 𝚫) for which $\left({\Delta \over a}\right)=+1$ , $\left({\Delta \over b}\right)=-1$ respectively. In the ideal theory the only way in which these formulae have been obtained is 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 Ω is expressed in the form [𝜔₁, 𝜔₂]; then any element of it is expressible as 𝑥𝜔₁＋𝑦𝜔₂, where 𝑥, 𝑦 are rational integers, and we shall have 𝖭 (𝑥𝜔₁＋𝑦𝜔₂)＝𝒂𝑥²＋𝒃𝑥𝑦＋𝒄𝑦², where 𝒂, 𝒃, 𝒄 are rational numbers contained in the ideal. If we put 𝑥＝𝛼𝑥′＋𝛽𝑦′, 𝑦＝𝛾𝑥′＋𝛿𝑦′, where 𝛼, 𝛽, 𝛾, 𝛿 are rational numbers such that 𝛼𝛿—𝛽𝛾＝±1, we shall have simultaneously (𝒂, 𝒃, 𝒄) (𝑥, 𝑦)²＝(𝒂′, 𝒃′, 𝒄′) (𝑥′, 𝑦′)² as in § 32 and also

(𝒂′,𝒃′,𝒄′) (𝑥′,𝑦′)²＝𝖭{𝑥′(𝛼𝜔₁＋𝛾𝜔₂)＋𝑦′(𝛽𝜔₁＋𝛿𝜔₂)}＝𝖭(𝑥′𝜔′₁＋𝑦′𝜔′₂),

where [𝜔′₁, 𝜔′₂] 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. Hence the class-number for ideals in Ω is also the class-number for a set of quadratic forms; and it can be shown that all these forms have the same determinant 𝚫. Conversely, every class of forms of determinant 𝚫 can be associated with a definite class of ideals in Ω(√𝑚), where 𝑚＝𝚫 or ⁠1/4⁠𝚫 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 𝒏: 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＋𝒏𝑖; and Hermite has considered bilinear forms 𝒂𝑥𝑥′＋𝒃𝑥𝑦′＋𝒃′𝑥′𝑦＋𝒄𝑦𝑦′, where 𝑥′, 𝑦′, 𝒃′ are the conjugates of 𝑥, 𝑦, 𝒃 and 𝒂, 𝒄, 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 Kronecker, the main principles of which are as follows. Let 𝖥( 𝑥, 𝑦, 𝑧, . . .) be a polynomial in any number of indeterminates (umbrae, as Sylvester calls them) with ordinary integral coefficients; if 𝒏 is the greatest common measure of the coefficients, we have 𝖥＝𝒏𝖤, 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 Ω of order 𝒏. Denoting the conjugate forms by 𝖥′, 𝖥″, . . . 𝖥^(𝒏－1), the product 𝖥𝖥′𝖥″ . . . 𝖥^(𝒏－1)＝𝑓𝖤, where 𝑓 is a real positive integer, and 𝖤 a unit form with real integral coefficients. The natural number 𝑓 is called the norm of 𝖥. If 𝖥, 𝖦 are any two forms (in Ω) we have 𝖭(𝖥𝖦)＝𝖭(𝖥)𝖭(𝖦). Let the coefficients of 𝖥 be 𝛼₁, 𝛼₂, those of 𝖦 𝛽₁, 𝛽₂, &c., and those of 𝖥𝖦 𝛾₁, 𝛾₂, &c.; and let 𝔭 be any prime ideal in Ω. Then if 𝔭^𝑚 is the highest power of 𝔭 contained in each of the coefficients 𝛼_𝑖, and 𝔭^𝒏 the highest power of 𝔭 contained in each of the coefficients 𝛽_𝑖, 𝔭^𝑚＋𝒏 is the highest power of 𝔭 contained by the whole set of coefficients 𝛾_𝑖. Writing dv(𝛼₁, 𝛼₂, . . .) for the highest ideal divisor of 𝛼₁, 𝛼₂, &c., this is called the content of 𝖥; and we have the theorem that the