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7h
comment Given $\frac{\sin(A - B)}{\sin(A+B)} = \frac57$, show $\tan A = 6 \tan B$
@user2650277 The first method is actually pretty simple. If you don't understand something, ask for clarification. Alonso is employing the implication $\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a+b}{a-b}=\frac{c+d}{c-d}$ using $a=\sin(A-B)$ and $b=\sin(A+B)$ (the expanded-out versions) and $c=5$, $d=7$.
1d
awarded  prime-numbers
1d
comment $3 X^3 + 4 Y^3 + 5 Z^3$ has roots in all $\mathbb{Q}_p$ and $\mathbb{R}$ but not in $\mathbb{Q}$
It helps us understand something beyond Hasse-Minkowski: it tells us the local-global philosophy is not as powerful (at least as we might initially conceive it) as an optimist would have wanted, because the logical extension from HM for quadratics to cubics fails. By the way, this polynomial is known as Selmer's example. See here for KCd's blurb on the topic.
Apr
13
comment What does inversion mean?
I don't know why you feel standard and unique "sound like opposites." Standard means something people customarily use all the time to the exclusion of alternatives. Something is unique if it is the only thing with a certain set of properties. These are the dictionary definitions of the words, more or less!
Apr
13
comment What does inversion mean?
It means things like "standard," "prescribed by an explicit rule," "natural and unique in a certain sense."
Apr
13
comment What does inversion mean?
Other examples might include normal, regular, perfect, simple, separable, elliptic, primitive, elementary, nice, complete (all of which I stole from the relevant MathOverflow thread).
Apr
13
comment What does inversion mean?
There are occasions when saying "additive inverse" is suggestive and useful, but dropping the "additive" part of the term should not be allowed if it could be confused with multiplicative inverse. When working at the elementary level (e.g. with rationals or reals) it's better still to just say "opposite," which is what I do and what I believe I see far more frequently. In any case, additive inverses, multiplicative inverses, and functional inverses are all special cases of inverses with respect to a binary operation - listing too many different usages of "inverse" can obfuscate the unity here.
Apr
13
comment Heuristic explanation for oscillatory behavior of first $n$ primes' multiples
Can you explain what your second graph is? What do the numbers on the $x$-axis represent? Are they multiples of $10^5$ or something? What is $n$ for it? I am not a programmer so I can't understand your code.
Apr
13
revised Heuristic explanation for oscillatory behavior of first $n$ primes' multiples
deleted 314 characters in body; edited tags; edited title
Apr
13
comment Subspace of $L(V)$
@GPerez What do you mean the statement for left and right ideals is false?
Apr
10
awarded  Popular Question
Apr
10
comment $a^2\equiv1 \pmod n$ iff $a\equiv\pm\,1\pmod p$ for all $p\mid n$
This does seem to be the "best-fit" interpretation of the OP - the closest thing to what OP was doing that is actually correct - but it strikes me as strange that OP would accidentally put "for all $p\mid n$" in the title; it's a rather sophisticated typo! I think we should await OP's clarification.
Apr
10
revised $a^2\equiv1 \pmod n$ iff $a\equiv\pm\,1\pmod p$ for all $p\mid n$
edited body
Apr
9
comment Prime ideals lying above in $\mathbb{Q}(\sqrt{-5})$
nice 'nym ${}{}$
Apr
6
comment Why do mathematicians study elementary abelian groups?
@Timbuc, yes, but just studying finite abelian groups in the abstract does not mean elementary abelian ones were picked out specifically and highlighted with a focus, any more than studying finite groups in the abstract means one studied groups of squarefree order as a special focus (did you know we have an explicit formula for the # of groups with any given sqrfree order? that doesn't make it into textbooks). | OP, pretty much any linear algebra you can do with an infinite vector space you can do with a finite one. Except you can't divide by the characteristic of the scalar field.
Apr
6
comment Why do mathematicians study elementary abelian groups?
I don't think elementary abelian groups are interesting in their own right (except if you want to go in the vector space direction), but in situations where groups arise we can sometimes study those situations which give arise specifically to elementary abelian groups - as an initial case study with tame ecomplexity, say. (E.g. as Galois groups or subfactors of another group.) @Timbuc Perhaps OP means they did not specifically highlight elementary abelian groups when covering structure theory of finite abelian groups.
Apr
6
comment Find all polynomials $P(x) \in \mathbb{Z}[X]$ such that $P(n) \mid 2^n-1 $
Oops yes, that was silly. The correct fact is $n,m>0$ coprime $\Rightarrow P(n),P(m)$ coprime.
Apr
5
comment Why is the argument of complex number determined up to integer multiple of $2 \pi$?
Well, travelling a distance of $\theta$ counterclockwise around the unit circle winds up in the same place as travelling $\theta+2\pi$ around, for instance. So the answer to the question "how far around have we travelled" is only determined up to additions and subtractions of $2\pi$.
Apr
4
comment Period of a module vs element
The principal ideal and its principal generator are sometimes interchanged. Is that actually causing you any problems?
Apr
4
comment Period of a module vs element
Yes it's defining $d$ to be the period of $M$. Is that your question?