Two centuries ago, in his celebrated work Disquisitiones Arithmeticae of 1801, Gauss laid down the beautiful law of composition of integral binary quadratic forms which would play such a critical role in number theory in the decades to follow. Even today, two centuries later, this law of composition still remains one of the primary tools for understanding and computing with the class groups of quadratic orders. It is hence only natural to ask whether higher analogues of this composition law exist that could shed light on the structure of other algebraic number rings and ﬁelds. ...
Annals of Mathematics
In our ﬁrst article  we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number ﬁelds, while the spaces underlying those composition laws were closely related to certain exceptional Lie groups.
In the ﬁrst three parts of this series, we considered quadratic, cubic and quartic rings (i.e., rings free of ranks 2, 3, and 4 over Z) respectively, and found that various algebraic structures involving these rings could be completely parametrized by the integer orbits of an appropriate group representation on a vector space. These orbit results are summarized in Table 1.
In the ﬁrst two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic ﬁelds could be explicitly parametrized. In particular, a central role in the theory was played by the parametrizations of the quadratic and cubic rings themselves. These parametrizations are beautiful and easy to state.
For the 50 industrial companies in the S&P index, the ratio of the total market capitalization
to net earnings is 18.4. Aggregate earnings and this price-earnings ratio imply an
estimate for the aggregate market capitalization of 1.62 times GNP. This is slightly higher
than Sloan’s (1936) estimate, which was based on a broader subset of industrial companies.