Hurwitz's theorem on sums of squares, proved in 1898 by Adolf Hurwitz, states that the multiplicative formula
\[
\left( \sum_{i=1}^{n} x_i^2 \right)\left( \sum_{j=1}^{n} y_j^2 \right) = \sum_{\ell=1}^{n} z_\ell^2,
\]
where \(x_1, \dots, x_n\) and \(y_1, \dots, y_n\) are arbitrary real numbers, and each component \(z_\ell\) is given as a linear combination of terms of the form \(x_i y_j\), namely
\[
z_\ell = \sum_{i=1}^{n} \sum_{j=1}^{n} A_{ij\ell}\, x_i y_j,
\]
with \(A_{ij\ell} \in \mathbb{R}\) constants independent of the choice of \(x\) and \(y\), holds only when \(n \in \{1,2,4,8\}\).
The aim of this thesis is to present a clear proof of this theorem that is accessible to undergraduate students of mathematics.
The thesis includes a survey of the history of research on sums of squares, with particular emphasis on the work of Diophantus, Euler, and Hamilton. We also discuss division algebras over the real numbers and present the Cayley–Dickson construction, which is crucial for understanding Hurwitz's theorem.
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