Page:EB1911 - Volume 19.djvu/888

 where $$h_1, h_2, \dots h_n$$ are elements of $$\mathrm{N}$$ which may be called the co-ordinates of $$\omega$$ with respect to the base $$\omega_1, \omega_2, \dots \omega_n$$. Thus $$\mathfrak{o}$$ is a modulus (&sect; 44), and we may write $$\mathfrak{o}=[\omega_1, \omega_2, \dots \omega_n]$$. Having found one base, we can construct any number of equivalent bases by means of equations such as $${\omega_i}' = \underset{i}{\Sigma} c_{ij}\omega_j$$, where the rational integral coefficients $$c_{ij}$$ are such that the determinant $$|c_{nn}| = \pm 1$$.

If we write $\surd\Delta = \begin{vmatrix} \omega_1, & \omega_2, & \dots & \omega_n \\ {\omega'}_1, & {\omega'}_2, & \dots & {\omega'}_n \\ {\omega}_1, & {\omega}_2, & \dots & {\omega''}_n \\ \vdots \\ {\omega^{(n-1)}}_1, & {\omega^{(n-1)}}_2, & \dots & {\omega^{(n-1)}}_n \\ \end{vmatrix}$|undefined

$$\Delta$$ is a rational integer called the discriminant of the field. Its value is the same whatever base is chosen.

If $$\alpha$$ is any integer in $$\Omega$$, the product of $$\alpha$$ and its conjugates is a rational integer called the norm of $$\alpha$$, and written $$\mathrm{N}(\alpha)$$. By considering the equation satisfied by $$\alpha$$ we see that $$\mathrm{N}(\alpha)=\alpha\alpha_1$$ where $$\alpha_1$$ is an integer in $$\Omega$$. It follows from the definition that if $$\alpha, \beta$$ are any two integers in $$\Omega$$, then $$\mathrm{N}(\alpha\beta) = \mathrm{N}(\alpha)\mathrm{N}(\beta)$$; and that for an ordinary real integer $$m$$, we have $$\mathrm{N}(m)=m^n$$.

 46. Ideals.—The extension of Kummer’s results to algebraic numbers in general was independently made by J. W. R. Dedekind and Kronecker; their methods differ mainly in matters of notation and machinery, each having special advantages of its own for particular purposes. Dedekind’s method is based upon the notion of an ideal, which is defined by the following properties:—

(i.) An ideal $$\mathfrak{m}$$ is an aggregate of integers in $$\Omega$$.

(ii.) This aggregate is a modulus; that is to say, if $$\mu, \mu'$$ are any two elements of $$\mathfrak{m}$$ (the same or different) $$\mu-\mu'$$ is contained in $$\mathfrak{m}$$. Hence also $$\mathfrak{m}$$ contains a zero element, and $$\mu+\mu'$$ is an element of $$\mathfrak{m}$$.

(iii.) If $$\mu$$ is any element of $$\mathfrak{m}$$, and $$\omega$$ any element of $$\mathfrak{o}$$, then $$\omega\mu$$ is an element of $$\mathfrak{m}$$. It is this property that makes the notion of an ideal more specific than that of a modulus.

It is clear that ideals exist; for instance, $$\mathfrak{o}$$ itself is an ideal. Again, all integers in $$\Omega$$ which are divisible by a given integer $$\alpha$$ (in $$\mathfrak{o}$$) form an ideal; this is called a principal ideal, and is denoted by $$\mathfrak{o}\alpha$$. Every ideal can be represented by a base (&sect;&sect; 44, 45), so that we may write $$\mathfrak{m} = [\mu_1, \mu_2, \dots \mu_n]$$, meaning that every element of $$\mathfrak{m}$$ can be uniquely expressed in the form $$\Sigma h_i\mu_i$$, where $$h_i$$ is a rational integer. In other words, every ideal has a base (and therefore, of course, an infinite number of bases). If $$\mathfrak{a}, \mathfrak{b}$$ are any two ideals, and if we form the aggregate of all products $$\alpha\beta$$ obtained by multiplying each element of the first ideal by each element of the second, then this aggregate, together with all sums of such products, is an ideal which is called the product of $$\mathfrak{a}$$ and $$\mathfrak{b}$$ and written $$\mathfrak{ab}$$ or $$\mathfrak{ba}$$. In particular $$\mathfrak{oa} = \mathfrak{a}, \mathfrak{o}^2 = \mathfrak{o}, \mathfrak{o}\alpha\ .\ \mathfrak{o}\beta = \mathfrak{o}\alpha\beta$$. This law of multiplication is associative as well as commutative. It is clear that every element of $$\mathfrak{ab}$$ is contained in $$\mathfrak{a}$$: it can be proved that, conversely, if every element of $$\mathfrak{c}$$ is contained in $$\mathfrak{a}$$, there exists an ideal $$\mathfrak{b}$$ such that $$\mathfrak{ab} = \mathfrak{c}$$. In particular, if $$\alpha$$ is any element of $$\mathfrak{a}$$, there is an ideal $$\mathfrak{a}'$$ such that $$\mathfrak{o}\alpha = \mathfrak{aa}'$$. A prime ideal is one which has no divisors except itself and $$\mathfrak{o}$$. It is a fundamental theorem that every ideal can be resolved into the product of a finite number of prime ideals, and that this resolution is unique. It is the decomposition of a principal ideal into the product of prime ideals that takes the place of the resolution of an integer into its prime factors in the ordinary theory. It may happen that all the ideals in $$\Omega$$ are principal ideals; in this case every resolution of an ideal into factors corresponds to the resolution of an integer into actual integral factors, and the introduction of ideals is unnecessary. But in every other case the introduction of ideals or some equivalent notion, is indispensable. When two ideals have been resolved into their prime factors, their greatest common measure and least common multiple are determined by the ordinary rules. Every ideal may be expressed (in an infinite number of ways) as the greatest common measure of two principal ideals.

47. There is a theory of congruences with respect to an ideal modulus. Thus $$\alpha \equiv \beta \pmod{\mathfrak{m}}$$ means that $$\alpha-\beta$$ is an element of $$\mathfrak{m}$$. With respect to $$\mathfrak{m}$$, all the integers in $$\Omega$$ may be arranged in a finite number of incongruent classes. The number of these classes is called the norm of $$\mathfrak{m}$$, and written $$\mathrm{N}(\mathfrak{m})$$. The norm of a prime ideal $$\mathfrak{p}$$ is some power of a real prime $$p$$; if $$\mathrm{N}(\mathfrak{p}) = p^f$$, $$\mathfrak{p}$$ is said to be a prime ideal of degree $$f$$. If $$\mathfrak{m}, \mathfrak{n}$$ are any two ideals, then $$\mathrm{N}(mn) =\mathrm{N}(\mathfrak{m})\mathrm{N}(\mathfrak{n})$$. If $$\mathrm{N}(\mathfrak{m}) = m$$, then $$m\equiv 0 \pmod{\mathfrak{m}}$$, and there is an ideal $$\mathfrak{m'}$$ such that $$\mathfrak{o}m = \mathfrak{mm}'$$. The norm of a principal ideal $$\mathfrak{o}\alpha$$ is equal to the absolute value of $$\mathrm{N}(\alpha)$$ as defined in &sect; 45.

The number of incongruent residues prime to $$\mathfrak{m}$$ is— $\phi(\mathfrak{m}) = \mathrm{N}(\mathfrak{m})\Pi\left(1-\frac{1}{\mathrm{N}(\mathfrak{p})}\right)$, where the product extends to all prime factors of $$\mathfrak{m}$$. If $$\omega$$ is any element of $$\mathfrak{o}$$ prime to $$\mathfrak{m}$$, $\omega^{\phi(\mathfrak{m})} \equiv 1 \pmod{\mathfrak{m}}$.|undefined

Associated with a prime modulus $$\mathfrak{p}$$ for which $$\mathrm{N}(\mathfrak{p})=p^f$$ we have $$\phi(p^f-1)$$ primitive roots, where $$\phi$$ has the meaning given to it in the ordinary theory. Hence follow the usual results about exponents, indices, solutions of linear congruences, and so on. For any modulus $$\mathfrak{m}$$ we have $$\mathrm{N}(\mathfrak{m})=\Sigma\phi(\mathfrak{d})$$, where the sum extends to all the divisors of $$\mathfrak{m}$$.

48. Every element of $$\mathfrak{o}$$ which is not contained in any other ideal is an algebraic unit. If the conjugate fields $$\Omega, \Omega', \dots \Omega^{}(n-1)$$ consist of $$r_1$$ real and $$2r_2$$ imaginary fields, there is a system of units $$\epsilon_1, \epsilon_2, \dots \epsilon_r$$, where $$r = r_1+r_2-1$$, such that every unit in $$\Omega$$ is expressible in the form $$\epsilon = \rho{\epsilon_1}^a{\epsilon_2}^b \dots {\epsilon_r}^l$$ where $$\rho$$ is a root of unity contained in $$\Omega$$ and $$a, b, \dots l$$ are natural numbers. This theorem is due to Dirichlet.

The norm of a unit is $$+1$$ or $$-1$$; and the determination of all the units contained in a given field is in fact the same as the solution of a Diophantine equation $\mathrm{F}(h_1, h_2, \dots h_n) = \pm 1$.

For a quadratic field the equation is of the form $${h_1}^2-n{h_2}^2 = \pm 1$$, and the theory of this is complete; but except for certain special cubic corpora little has been done towards solving the important problem of assigning a definite process by which, for a given field, a system of fundamental units may be calculated. The researches of Jacobi, Hermite, and Minkowsky seem to show that a proper extension of the method of continued fractions is necessary.

 49. Ideal Classes.—If $$\mathfrak{m}$$ is any ideal, another ideal $$\mathfrak{n}$$ can always be found such that $$\mathfrak{mn}$$ is a principal ideal; for instance, one such multiplier is $$\mathfrak{m}^{-1}\mathrm{N}(\mathfrak{m})$$. Two ideals $$\mathfrak{m}, \mathfrak{m}'$$ are said to be equivalent ($\mathfrak{m}\sim\mathfrak{m}'$) or to belong to the same class, if there is an ideal $$\mathfrak{n}$$ such that $$\mathfrak{mn}, \mathfrak{m'n}$$ are both principal ideals. It can be proved that two ideals each equivalent to a third are equivalent to each other and that all ideals in $$\Omega$$ may be distributed into a finite number, $$h$$, of ideal classes. The class which contains all principal ideals is called the principal class and denoted by $$\mathrm{O}$$.

If $$\mathfrak{m}, \mathfrak{n}$$ are any two ideals belonging to the classes $$\mathrm{A}, \mathrm{B}$$ respectively, then $$\mathfrak{mn}$$ belongs to a definite class which depends only upon $$\mathrm{A}, \mathrm{B}$$ and may be denoted by $$\mathrm{AB}$$ or $$\mathrm{BA}$$ indifferently. Thus the class-symbols form an Abelian group of order $$h$$, of which $$\mathrm{O}$$ is the unit element; and, mutatis mutandis, the theorems of &sect; 37 about composition of classes still hold good.

The principal theorem with regard to the determination of $$h$$ is the following, which is Dedekind’s generalization of the corresponding one for quadratic fields, first obtained by Dirichlet. Let $\zeta(s) = \underset{(\mathfrak{m})}{\Sigma}\mathrm{N}(\mathfrak{m})-^s $ where the sum extends to all ideals $$\mathfrak{m}$$ contained in $$\Omega$$; this converges so long as the real quantity $$s$$ is positive and greater than $$1$$. Then $$\kappa$$ being a certain quantity which can be calculated when a fundamental system of units is known, we shall have $\kappa h = \underset{s=1}{Lt}\{(s-1)\zeta(s)\}$.

The expression for $$\kappa$$ is rather complicated, and very peculiar; it may be written in the form $\kappa = \frac{2^{r_1+r_2}\pi^{r_2}}{w}.\frac{\mathrm{R}}{|\surd\Delta|}$|undefined where $$|\surd\Delta|$$ means the absolute value of the square root of the discriminant of the field, $$r_1, r_2$$ have the same meaning as in &sect; 48, $$w$$ is the number of roots of unity in $$\Omega$$, and $$\mathrm{R}$$ is a determinant of the form $$|l_i(\epsilon^j)|$$, of order $$(r_1+r_2-1)$$, with elements which are, in a certain special sense, “logarithms” of the fundamental units $$\epsilon_1, \epsilon_2, \dots \epsilon_r$$.

50. The discriminant $$\Delta$$ enjoys some very remarkable properties. Its value is always different from $$\pm 1$$; there can be only a finite number of fields which have a given discriminant; and the rational prime factors of $$\Delta(\Omega)$$ are precisely those rational primes which, in $$\Omega$$, are divisible by the square (or some higher power) of a prime ideal. Consequently, every rational prime not contained in $$\Delta$$ is resolvable, in $$\Omega$$, into the product of distinct primes, each of which occurs only once. The presence of multiple prime factors in the discriminant was the principal difficulty in the way of extending Kummer’s method to all fields, and was overcome by the introduction of compound moduli—for this is the common characteristic of Dedekind’s and Kronecker’s procedure.

 51. Normal Fields.&mdash;The special properties of a particular field $$\Omega$$ are closely connected with its relations to the conjugate fields $$\Omega', \Omega, \dots \Omega^{(n-1)})$$. The most important case is when each of the conjugate fields is identical with $$\Omega$$: the field is then said to be Galoisian or normal''. The aggregate $$\mathrm{R}(\Omega, \Omega', \dots \Omega^{(n-1)}))$$ of all rational functions of $$\theta$$ and its conjugates is a normal field: hence every arithmetical field of order $$n$$ is either normal, or contained in a normal field of a higher order. The roots of an equation $$f(\theta)=0$$ which defines a normal field are associated with a group of substitutions: if this is Abelian, the field is called Abelian; if it is cyclic, the field is called cyclic. A cyclotomic field is one the elements of which are all expressible as rational functions of roots of unity; in particular the complete cyclotomic field $$\mathrm{C}_m$$, of order $$\phi(m)$$, is the aggregate of all rational functions of a primitive mth root of unity. To Kronecker is due the very remarkable theorem that all Abelian (including cyclic) fields are cyclotomic: the first published proof of this was given by Weber, and another is due to D. Hubert.

Many important theorems concerning a normal field have been established by Hilbert. He shows that if $$\Omega$$ is a given normal field of order $$m$$, and $$\mathfrak{p}$$ any of its prime ideals, there is a finite series of associated fields $$\Omega_1, \Omega_2, \!\And\!\!\!\!\mathrm{c}.$$, of orders $$m_1, m_2, \!\And\!\!\!\!\mathrm{c}.$$, such that $$m_i \equiv 0 \pmod{m_{i+1}}$$, and that if $$r_i = m/m_i$$, $$\mathfrak{p}^{r_i} = \mathfrak{p}_i$$, a prime ideal in $$\Omega_i$$. If $$\Omega_l$$ is the last of this series, it is called the field of inertia