20.500.12556/RUL-172128 Hurwitzov izrek o vsoti kvadratov delo diplomskega seminarja Hurwitz’s theorem on sums of squares Hurwitzov izrek o vsoti kvadratov, ki ga je leta 1898 dokazal Adolf Hurwitz, pravi, da multiplikativna formula: \[ \left( \sum_{i=1}^{n} x_i^2 \right) \cdot \left(\sum_{j=1}^{n} y_j^2\right) = \sum_{l=1}^{n} z_l^2, \] kjer so \( x_1, \dots, x_n \) in \( y_1, \dots, y_n\) poljubna realna števila, vsaka komponenta \( z_l \) pa je dana kot linearna kombinacija izrazov oblike \( x_i y_j \), torej: \[ z_l = \sum_{i=1}^{n} \sum_{j=1}^{n} A_{ijl} \cdot x_i y_j, \] pri čemer so \( A_{ijl} \in \mathbb{R} \) konstante, neodvisne od izbire \( x \) in \( y \), velja le, če je \( n = 1, 2, 4 \) ali \(8\). Cilj diplomske naloge je predstaviti jasen dokaz tega izreka, ki je primeren tudi za dodiplomske študente matematike. Naloga vključuje pregled zgodovine raziskav na področju vsot kvadratov, s posebnim poudarkom na delu Diofanta, Eulerja in Hamiltona. Obravnavamo tudi algebre z deljenjem nad realnimi števili in predstavimo Cayley-Dicksonovo konstrukcijo, ki je ključna za razumevanje Hurwitzovega izreka. 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. algebra matrika kvadrat vsota Hurwitzov izrek algebra matrix square sum Hurwitz’s theorem true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2025-09-06 08:15:21 2025-09-06 08:15:24 2025-10-06 03:56:14 0000-00-00 00:00:00 2025 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 511 152692 248059907 19208.pdf 19208.pdf 1 F7F5B15ECFF35852440EC2F20B72191C aa19dc5dc2a35fb7f91aa701fecc96ff503b6e024a5be41233f03c5ddf473104 b961a053-8ae8-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=215928 Fakulteta za matematiko in fiziko 0 0 0