1911 Encyclopædia Britannica/Number/Ideals

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 (§§ 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 § 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 ${\mathfrak {m}}$,

$\omega ^{\phi ({\mathfrak {m}})}\equiv 1{\pmod {\mathfrak {m}}}$.

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.