Men of Mathematics (91 page)

Another concept, that of an
algebraic number field
of
degree n
is now introduced: if r is an algebraic number of degree
n,
the totality of all expressions that can be constructed from
r
by repeated additions, subtractions, multiplications, and divisions (division by zero is not defined and hence is not attempted or permitted), is called
the algebraic number field generated by
r, and may be denoted by
F[r]
. For example, from
r we get r +
r,
or 2r; from this and r we get 2r/r or 2, 2r—r or r, 2r × r or 2r
²
, etc. The
degree
of this
F[r]
is w.

It can be proved that every member of
F[r]
is of the form
c
₀
r
^n-
l
+
c
₁
r
^n-2
+ . . .+
c
_n-1
, where the c's are rational numbers, and further every member of F[r]
is an algebraic number of degree not greater than
n
(in fact the degree is some divisor of
n). Some,
but not all, algebraic numbers in
F[r]
will be algebraic
integers.

The central problem of the theory of algebraic numbers is to investigate the laws of arithmetical divisibility of algebraic integers in an algebraic number field of degree
n.
To make this problem definite it is necessary to lay down exactly what is meant by “arithmetical divisibility,” and for this we must understand the like for the
rational
integers.

We say that one rational integer,
m,
is divisible by another,
d,
if we can find a rational integer,
q,
such that
m = q
×
d; d
(also
q)
is called a
divisor
of
m.
For example 6 is a divisor of 12, because 12 = 2 ×
6; 5
is not a divisor of 12 because there does not exist a rational integer
q
such that 12 = q × 5.

A (positive) rational
prime
is a rational integer greater than 1 whose only positive divisors are 1 and the integer itself. When we try to extend this definition to algebraic integers we soon see that we have not found the root of the matter, and we must seek some property of rational primes which can be carried over to algebraic integers. This property is the following: if a rational prime
p
divides the product
a
×
b
of two rational integers, then (it can be proved that)
p
divides at least one of the factors
a, b
of the product.

Considering the unit, 1, of rational arithmetic, we notice that 1 has the peculiar property that it divides
every
rational integer; −1 also has the same property, and 1,-1 are the
only
rational integers having this property.

These and other clues suggest something simple that will work, and we lay down the following definitions as the basis for a theory of arithmetical divisibility for algebraic integers. We shall suppose that all the integers considered lie in an algebraic number field of degree
n.

If
r, s, t
are algebraic integers such that r =
s
×
t,
each of
s, t
is called a
divisor
of
r.

If
j
is an algebraic integer which divides
every
algebraic integer in the field,
j
is called a
unit
(in that field). A given field may contain an
infinity of units, in distinction to the pair 1, −1 for the rational field, and this is one of the things that breeds difficulties.

The next introduces a radical and disturbing distinction between rational integers and algebraic integers of degree greater than 1.

An algebraic integer other than a unit whose only divisors are units and the integer itself, is called
irreducible.
An irreducible algebraic integer which has the property that
if
it divides the product of two algebraic integers,
then
it divides at least one of the factors, is called a
prime
algebraic integer. All primes are irreducibles, but not all irreducibles are primes in
some
algebraic number fields, for example in
as will be seen in a moment. In the common arithmetic of 1, 2, 3 . . . the irreducibles and the primes are the same.

In the chapter on Fermat the fundamental theorem of (rational) arithmetic was mentioned: a rational integer is the product of (rational) primes
in only one way.
From this theorem springs all the intricate theory of divisibility for rational integers. Unfortunately the fundamental theorem does
not
hold in
all
algebraic number fields of degree greater than one, and the result is chaos.

To give an instance (it is the stock example usually exhibited in text books on the subject), in the field
we have

each of
is a prime in this field (as may be verified with some ingenuity), so that 6, in this field, is
not uniquely
decomposable into a product of primes.

It may be stated here that Kronecker overcame this difficulty by a beautiful method which is too detailed to be explained untechnically, and that Dedekind did likewise by a totally different method which is much easier to grasp, and which will be noted when we consider his life. Dedekind's method is the one in widest use today, but this does not imply that Kronecker's is less powerful, nor that it will not come into favor when more arithmeticians become familiar with it.

*  *  *

In his dissertation of 1845 Kronecker attacked the theory of the units in certain special fields—those defined by the equations arising from the algebraic formulation of Gauss' problem to divide the circumference of a circle into
n
equal parts or, what is the same, to construct a regular polygon of
n
sides.

We can now close up one part of the account opened by Fermat.

In struggling to prove Fermat's “Last Theorem” that
x
ⁿ
+
y
ⁿ
= z
ⁿ
is impossible in rational integers
x, y, z
(none zero) if
n
is an integer greater than 2, arithmeticians took what looks like a natural step and resolved the left-hand side,
x
ⁿ
+ y
ⁿ
,
into its
n
factors of the first degree (as is done in the usual second course of school algebra). This led to the exhaustive investigation of the algebraic number field mentioned above in connection with Gauss' problem—after serious but readily understandable mistakes had been made.

The problem at first was studded with pitfalls, into which many a competent mathematician and at least one great one—Cauchy—tumbled headlong. Cauchy assumed as a matter of course that in the algebraic number field concerned the fundamental theorem of arithmetic must hold. After several exciting but premature communications to the French Academy of Sciences, he admitted his error. Being restlessly interested in a large number of other problems at the time, Cauchy turned aside and failed to make the great discovery which was well within the capabilities of his prolific genius and left the field to Kummer. The central difficulty was serious: here was a species of “integers”—those of the field concerned—which defied the fundamental theorem of arithmetic; how reduce them to law and order?

The solution of this problem by the invention of a totally new kind of “number” appropriate to the situation, which (in terms of these “numbers”) automatically restored the fundamental theorem of arithmetic, ranks with the creation of non-Euclidean geometry as one of the outstanding scientific achievements of the nineteenth century, and it is well up in the high mathematical achievements of all history. The creation of the new “numbers”—so-called “ideal numbers”—was the invention of Kummer in
1845.
These new “numbers” were not constructed for all algebraic number fields but only for those fields arising from the division of the circle.

Kummer too had fallen afoul of the net which snared Cauchy, and for a time he believed that he had proved Fermat's “Last Theorem.” Then Dirichlet, to whom the supposed proof was submitted for criticism, pointed out by means of an example that the fundamental theorem of arithmetic, contrary to Kummer's tacit assumption, does
not
hold in the field concerned. This failure of Kummer's was one of the most fortunate things that ever happened in mathematics. Like Abel's initial mistake in the matter of the general quintic, Kummer's turned him into the right track, and he invented his “ideal numbers.”

Kummer, Kronecker, and Dedekind in their invention of the modern theory of algebraic numbers, by enlarging the scope of arithmetic
ad infinitum
and bringing algebraic equations within the purview of number, did for the higher arithmetic and the theory of algebraic equations what Gauss, Lobatchewsky, Johann Bolyai, and Riemann did for geometry in emancipating it from slavery in Euclid's too narrow economy. And just as the inventors of non-Euclidean geometry revealed vast and hitherto unsuspected horizons to geometry and physical science, so the creators of the theory of algebraic numbers uncovered an entirely new light, illuminating the whole of arithmetic and throwing the theories of equations, of systems of algebraic curves and surfaces, and the very nature of number itself, into sharp relief against a firm background of shiningly simple postulates.

The creation of “ideals”—Dedekind's inspiration from Kummer's vision of “ideal numbers”—renovated not only arithmetic but the whole of the algebra which springs from the theory of algebraic equations and systems of such equations, and it proved also a reliable clue to the inner significance of the “enumerative geometry”
^I
of Plücker, Cayley and others, which absorbed so large a fraction of the energies of the geometers of the nineteenth century who busied themselves with the intersections of nets of curves and surfaces. And last, if Kronecker's heresy against Weierstrassian analysis (noted later) is some day to become a stale orthodoxy, as all not utterly insane heresies sooner or later do, these renovations of our familiar integers,
1, 2, 3,
 . . . , on which all analysis strives to base itself, may ultimately indicate extensions of analysis, and the Pythagorean speculation may envisage generative properties of “number” that Pythagoras never dreamed of in all his wild philosophy.