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6h
comment How does one prove that $\mathbb{Z}[\sqrt{-41}]$ is a unique factorization domain.
Something is off here. I think you are referring to discriminant $-163,$ the principal (and only class) form of that discriminant is $f(x,y) = x^2 + xy+41 y^2.$
7h
revised Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0
added 539 characters in body
9h
comment Squares in $\mathbb Z_p$
I like the treatment of piadic numbers in Cassels, Rational Quadratic Forms, tables about your question approximately page 40-43. store.doverpublications.com/0486466701.html
9h
comment Squares in $\mathbb Z_p$
what you have written in the first sentence is false.
11h
revised How to improve visualization skills (Graphing)
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11h
revised What exactly is a Diophantine representation?
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11h
answered What exactly is a Diophantine representation?
12h
comment What exactly is a Diophantine representation?
math.umd.edu/~laskow/Pubs/713/Diorepofprimes.pdf
12h
answered How to improve visualization skills (Graphing)
14h
comment Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0
@copper.hat, that worked out. I should read some of my books, on Witt's Lemma, and see how the final step happens without guessing at factoring polynomials.
14h
revised Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0
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15h
comment Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0
@copper.hat, right, no need for eigenvalues, the null cone of a quadratic form is not the same as the nullspace of its Gram/Hessian matrix. Now I need to think of how to extract a vector space in the null cone, never done that before.
15h
answered Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0
15h
comment Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0
This is Witt cancellation, it is not necessary to find eigenvectors. Reduction of the quadratic form in the general method of Hermite suffices. Why do you say this is an exam question?
1d
comment How many real numbers satisfy the following
$5 \pi/2 \approx 7.85,$ so doing $\sin x = \frac{x}{8}$ for $-8 \leq x \leq 8$ should give a good idea of what is going on.
1d
comment How many real numbers satisfy the following
try things like $\sin x = \frac{x}{3}$ first, where you can draw the graphs of $y = \sin x$ and $y = \frac{x}{3}$ yourself. graph paper, five lines per inch: printablepaper.net/preview/grid-portrait-letter-5-index Valuable skill, drawing graphs
1d
comment Roadmap to $p$-adic numbers: where a self-learner should look for references
Rational Quadratic Forms by Cassels. The most satisfying description on Hasse-Minkowski, along with how to calculate things. After that, if you learn how to use the Mass Formula for a genus of (positive) forms, and the representation counts for a genus representing numbers by Siegel's formula, you will be ahead of the game. I contiue to think that I would have made far more sense of Lie algebras if I had had quadratic forms first, perhaps in the guise of integral lattices.
1d
comment How to use TI-Nspire CX CAS to solve Diophantine equation?
You can give somebody your TI-Nspire in exchange for the answer.
1d
comment Product of two integers of the form $x^2+my^2$ is of the same form.
Cox, especially page 49 wiley.com/WileyCDA/WileyTitle/productCd-1118390180.html
1d
comment Product of two integers of the form $x^2+my^2$ is of the same form.
This is the earliest case of Gauss composition of binary quadratic forms; you can look this up in various places, such as the book by Cox, Primes of the Form $x^2 + n y^2.$ The version of this that we use is actually due to Dirichlet, with the phrase "united forms" of importance; it turns out that Gauss's long description of what a composition operation should accomplish can be satisfied by 14 different operations. This observation is due to Bhargava.