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Theorem 3.9

I am self taught on this topic and I don't know if I'm being stupid but what is meant to be in place of the box in the statement for theorem 3.9?

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    It means a square. Not in geometry, like something of the form $x^2$. Hopefully he notes this at some point earlier in the text, but if not it's still pretty common. – 2017-01-04
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    I can't find a definition for it earlier in the text, is there a formal definition for it? – 2017-01-04
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    Yes, anything of the form $x^2$. It's common among many mathematicians to avoid writing cumbersome sentences (because we are lazy) such as "let $y=x^2$ be a square" with "Let $a=\square$" similarly for non-squares. We do it because we don't care what it's the square of--the notation $y=x^2$ might make $x$ relevant at some point, the main point is that it is the square of **something**, we don't actually care about $x$ [poor $x$ :-( ] – 2017-01-04
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    Okay thank you, that helps a lot! – 2017-01-04
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    My pleasure, enjoy your studies: Quaternion algebras are a fun topic! – 2017-01-04
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    @AdamHughes It is common? I'm just surprised I've never seen it. Do you know of any authors that use it, offhand? – 2017-01-04
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    @rschwieb I don't know that it's very common in **books**: I see it a lot during number theory lectures, especially for people who study things where squares are very relevant (i.e. class field theory, quaternion algebras, and adjacent subjects). This is one of the reasons I was somewhat surprised there was no reference to the symbol earlier in the text (albeit the op may just have missed it). If the op will comment on which book it is one of us might also be able to confirm/repudiate this. In fact, before I saw this I thought it too informal to be used in books, but it must be catching on. – 2017-01-04
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    @AdamHughes If I would have seen it, I would have thought it was a latex typesetting error. Frankly, I think it's an awful choice. I can appreciate avoidance of cumbersome sentences, but really "such that $a$ is not a square mod $p$" is perfectly clear without being cumbersome, really. I guess maybe if you're using the phrase in every theorem you might adopt it successfully. – 2017-01-04
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    @rschwieb yes it's extremely common as a hypothesis. I don't know that I have seen it outside my area, but of course that's to be expected just by availability and culture differences. I also agree it's not a good choice for a book as it's not at all standard enough for that yet (unless previously defined in that book) – 2017-01-04

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