Claim: \int\frac{dx}{\sqrt{x}(1-x)}=\log{\frac{1+\sqrt{x}}{1-\sqrt{x}}}
Derivation confirms this, but how was this answer arrived at? IBP seems not to work, can't find a good u-substitution...
A book I'm reading (Companion to Concrete Math Vol. I by Melzak) mentions, "...any ellipse occurs as a plane section of any given cone. This is not the case with hyperbolas: for a fixed cone only those hyperbolas whose asymptotes make a sufficiently small angle occur as plane sections."
It...
Does anyone know how Madhava discovered the power series for the arctangent? I think the standard way is to note that 1-x^2+x^4-\dotsb converges uniformly on (-1,1) to \frac{d}{dt}\tan^{-1}x, and thus applying the fundamental theorem of calculus we may integrate term-by-term. But how did...
Suppose E and D are both finite extensions of F, with K being the Galois closure of \langle D,E \rangle (is this the correct way to say it?) Is it correct that E and D are conjugate fields over F iff G,H are conjugate subgroups, where G,H\leqslant \text{Aut}(K/F) are the subgroups which fix...
Where have you used that S\cap \langle B_i\rangle \neq \{0\} for all i? I can come up with the following counterexample if we do not assume this hypothesis:
In \mathbb{R}^2, the subspace y=x is certainly not the direct sum of its intersections with \langle e_1 \rangle and \langle e_2 \rangle...
I've been working on this Linear Algebra problem for a while: Let F be a field, V a vector space over F with basis \mathcal{B}=\{b_i\mid i\in I\}. Let S be a subspace of V, and let \{B_1, \dotsc, B_k\} be a partition of \mathcal{B}. Suppose that S\cap \langle B_i\rangle\neq \{0\} for all i...
A problem asks to find an abelian group V and a field F such that there exist two different actions, call them \cdot and \odot, of F on V such that V is an F-module.
A usual way to solve this is to take any vector space over a field with a non-trivial automorphism group, and define r\odot \mu...
OK, I agree that an algebraic extension is a union of finite extensions. I'm not quite seeing how that allows us to argue as if K/F is finite. (Perhaps I'm misunderstanding.)
Definitely--that is explicitly stated in the article. I was under the impression that separable implied algebraic...
I'm reading the following article by Maxwell Rosenlicht:
http://www.jstor.org/stable/2318066
(The question should be clear without the article, but I present it here for reference.)
In the beginning of the article he discusses differential fields (i.e. a field F with a map F\to F...
OK, how about this? Over F we have \alpha as the only root of f(x) (with multiplicity p). Let f(x)=p_1(x)\dotsb p_n(x) be a factorization of f(x) in K[x] into monic irreducibles. Then each of these must be the minimal polynomial of \alpha over K. So they all must have the same degree. So...