230 A History ofMathematics
While Ramanujan had to receive an accelerated schooling in what then counted as up-to-date
number theory, and much else, from Littlewood and Hardy, Emmy Noether was in a much more
usual European situation; trained in a rather old-fashioned approach to a central topic—crudely,
the relation of algebra to geometry—she came into contact with Hilbert and others who were in
the process of transforming it, abandoned her earlier lines of work and took the ideas which she
had received so much further as to have a crucial influence on the next generation. It would be
easy to point out how far she was from receiving the kind of recognition which a man would have
received in her career, and being a Jew and a Communist did not help in 1920s Germany. Forced
out of her untenured position at Göttingen by the Nazis, she found refuge at the women’s college
of Bryn Mawr, and by the time of her death had already made a decisive impact on the course of
mathematical history. Again, the central ideas are not easy to explain; however, one should try since
they are so much at the centre of what happened in twentieth-century mathematics (and at least
they are easier than those of Ramanujan). Partly they came from editing Dedekind’s posthumous
papers, partly from current work on polynomials, but she unified the two into what, as a result
of her work has become known as the general theory of ‘rings’ and ‘ideals’, the latter defined
by Dedekind. [Aring Ris a set in which addition and multiplication can be defined, satisfying
(to simplify) the same sensible rules as they do for integers; a subsetI ⊂Ris called anidealif
(a)a+b∈Iwhenaandb∈I, and (b)ab∈Iwhena∈I(not necessarilyb). Example: the set
of all multiples of 6 inZis an ideal, usually written (6).] These ideas entered into two apparently
disparate areas of mathematics.- In algebraic geometry, it had become common to consider not so much the curve C (for example)
defined by an equation likea^2 x^2 −b^2 y^2 − 1 =0 (Fig. 10), but theringof all polynomial
functions inxandy; the functions which vanished on C were then an ideal. Similarly for
algebraic manifolds (varieties) of higher dimensions. Instead of studying the geometric object,
(it came to be realized) one could equally well study the ideal. - In number theory, one routinely considered ‘number rings’, such as the set of allm+n
√
−5,
wheremandnare integers. It was to study these, their sometimes strange division and
factorization properties, and so to solve arithmetical problems that Kummer and Dedekind had
introduced ideals, or ‘ideal numbers’ in the first place.What Noether observed is clear from the above description—but only because the description
has been framed in the terms which she devised: that these two families of questions were both
concerned with the structure of ideals in a particular type of ring (it is now called ‘Noetherian’).6
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0
–2
–4
–6
Fig. 10Curve (hyperbola) with equation as in text;a= 1 /5,b= 1 /3.