It says - Numbers of the form a^2 + n*b^2 are closed under multiplication.
In more advanced language: For R a commutative ring (like say, the integers) the following function f is a ring homomorphism
f:R[√(-n)] -> M_2(R),
f(a + b√(-n)) = [[a, (-n)b], [b, a]]
coderatlarge•6mo ago
gjm11•6mo ago
It seems to be taken directly from the article, but it doesn't begin at the beginning or end at the end or shed any light on the part of the title most likely to be puzzling ("beyond arithmetic").
If the intention was to help out readers who don't know what a quadratic form is, I think a more helpful piece of advice would be: if you don't already know what a quadratic form is, then it is very unlikely that you will get anything much out of this article.
coderatlarge•6mo ago
i went down to Euler’s sums of two squares identity which i think demonstrates clearly that even middle school algebra suffices to get a sense of depth from this work showing that a product can also be seen as a sum in a more sophisticated context (ie beyond arithmetic structure emerges).
i fail to understand your disapproval on my providing additional context from the page to indicate that this is likely a more interesting post to a wider set of readers than most might assume just from the title. i didn’t feel the need to editorialize further because i thought the quote says it all on its own.
Jtsummers•6mo ago
Without that contextual clue, your comment appears to just be noise.
coderatlarge•6mo ago
pavel_lishin•6mo ago
But this isn't additional context. It's just a few copy-pasted paragraphs.
coderatlarge•6mo ago