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 5h comment Finding the widest angle to shoot a soccer ball from the sideline using optimization It might be good to point out how to check it truly is a maximum rather than say "it's apparent...". A striking example of a failure to verify that a solution to $f' = 0$ truly is a maximum rather than a minimum is at vozderazon.tripod.com/lit/stealthbomber.html and pages.iu.edu/~dpthurst/F10/CalcIII/StealthBomber.pdf. 5h comment Why the radius of convergence and not “areas of convergence” for power series? In fact, set theory was created by Cantor in response to questions about the domain of convergence of Fourier series, which can be a set that is a lot nastier than an interval. The domain of convergence of a power series (on the real line) is always an interval. 5h comment Can one define a functional on a Hilbert space based on its action on a Hilbert basis? You need a continuity condition as well. Simply assigning any random values you want to the Hilbert space basis is not going to extend to a continuous linear map. Do you believe that two (continuous) linear functionals that are equal on a Hilbert space basis are equal everywhere? 5h comment Research Paper on Math I realized that, but since you ultimately need to be writing a paper I wanted to point out that describing an equation as American or non-American is simply bad writing. Always be attentive to how you express your ideas. 5h comment differentiability, complex analysis Try working with some concrete examples (polynomials, rational functions) to see what a likely formula for the derivative of $f^*(z)$ should be. I personally think this is a really good problem. 5h comment If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$. Here is a concrete example: let $R = R' = {\mathbf Z}/(6)$. Let $f(x) = 3x$. Then $f(x)$ is additive and multiplicative (since $3^2 \equiv 3 \bmod 6$), but $f(1) \not= 1$. Note $f$ is not surjective. You should be made aware that the standard convention of most mathematicians who use algebra in their work (esp. anyone who uses commutative algebra) is that by definition a ring homomorphism must satisfy $f(1) = 1$. That is, most mathematicians who use algebra would not call a general additive/multiplicative map between rings a ring homomorphism unless it satisfies $f(1) = 1$. 6h comment Research Paper on Math Calling equations "non-American" makes me cringe. An equation does not have a nationality. (And you might want to read up on the journal Deutsche Mathemtatik.) Please watch how you express your ideas when you get around to writing the paper itself. 6h comment Composition of Polynomials and Galois Theory A Google search on "composition polynomials galois" returned the article ccms.or.kr/data/pdfpaper/jcms22_3/22_3_497.pdf as one of the top hits (did you try an internet search before asking your question?) and the paragraph after Lemma 3.2 will be of interest to you. 6h comment Is Euclidean geometry really a “dead” subject? If so, why? Your general premise is wrong. My university has a Euclidean and non-Euclidean geometry course (taken mostly by future high school math teachers). You say Harvard doesn't seem to have such a course, but did you see their Math 130? Though that is not a course aimed at the superior math majors. It is not exciting math compared with all the other topics that a math major could study (differential geometry, complex analysis, ...) 6h comment Is Euclidean geometry really a “dead” subject? If so, why? I think the question is still pretty rant-y. Euclidean geometry is not isolated. It extends to higher dimensions and in different directions (contrasts with non-Euclidean geometry). Math is a verrrrry big subject, and just as nobody learns an algorithm for computing cube roots by hand anymore (I barely got a glimpse of that when I was in high school) other material taught for a long time will have to give way so that more contemporary topics can be discussed. Latin and Greek used to be a standard part of everyone's higher education, but now.... 6h comment Provide an example to show that $S$ may not necessarily be a unique factorisation domain when $R$ is a unique factorisation domain. Did you even try to check your map $\mathbf Z[i] \rightarrow \mathbf Z[\sqrt{-5}]$ is a ring homomorphism? A homomorphism is not just symbol-pushing. 1d comment Restriction of topological ring isomorphism Since you speak categorically, first one needs to know that $R^\times$ belongs to the category of topological groups! 1d comment Restriction of topological ring isomorphism The previous comment focuses only on the topology, but the group structure alongside it is also important for your question to have meaning. Could you please tell us what you think the topology is on $R^\times$ that makes it a topological group? 1d comment prove that $x^2 + 5 =y^3$ has no solutions for $x,\ y \in \mathbb{Z}$ Do you know about ideal class groups? The ring $\mathbf Z[\sqrt{-5}]$ has class number $2$, which is relatively prime to the exponent $3$ in the equation, so even though $\mathbf Z[\sqrt{-5}]$ is not a UFD one can prove that $x+\sqrt{-5}$ must be a cube in $\mathbf Z[\sqrt{-5}]$, which is the kind of conclusion you'd expect if $\mathbf Z[\sqrt{-5}]$ were a UFD. Trying to solve $x + \sqrt{-5} = (a + b\sqrt{-5})^3$ in integers $a$ and $b$ leads to a contradiction by looking at the coefficient of $\sqrt{-5}$ on both sides. That's how you solve this with algebraic number theory. 1d comment prove that $x^2 + 5 =y^3$ has no solutions for $x,\ y \in \mathbb{Z}$ When you say $y = 1$ from $y^6 = 1$, actually you can only say $y = \pm 1$. But you still get a contradiction after that, as before. 1d comment An approach to proving that $\Bbb{Q}[x,y]/(x^3-y^2)$ is isomorphic to $\Bbb{Q}[t^2,t^3]$ It means nothing to "consider" them as separate variables. Evaluation at $(t^2,t^3)$ is a ring homomorphism $\mathbf Q[x,y] \rightarrow \mathbf Q[t^2,t^3]$ that is obviously surjective. Prove the kernel is $(x^3-y^2)$, i.e., for a polynomial $f(x,y)$ in $\mathbf Q[x,y]$ you have $f(t^2,t^3) = 0$ only if $f(x,y)$ is divisible by $x^3-y^2$. Note that your list of relations $x^{3n}=y^{2n}$ is woefully incomplete; those are just the easiest relations. They are definitely not all of them. The point of computing the kernel of that map is to show all relations are consequences of $x^3=y^2$. Apr15 comment Power series in p-adic integers This is discussed in nearly every book on $p$-adic analysis. Have you read about convergence of power series in any book on $p$-adics (Gouvea, Koblitz, Robert, ...)? Apr15 comment When is the normality of field extensions transitive? The field $\mathbf Q(\sqrt{2},\sqrt[3]{2},\zeta_3)$ is normal over $\mathbf Q$. It is a splitting field for $(x^2-2)(x^3-2)$ over $\mathbf Q$. You need to completely review this material, since you do not seem to have a strong base of examples that you understand well. If you are taking a class on Galois theory then have a talk with your instructor. Apr15 comment When is the normality of field extensions transitive? There is no absolute concept of splitting field. The whole idea is relative. You speak about the splitting field of a polynomial over a given field. And any two splitting fields of a chosen polynomial over a given field are isomorphic to each other by an isomorphism fixing that given field. You have a mistaken idea of what a splitting field means. Apr15 comment Show that the Euclidean algorithm works for Gaussian integers. Did you try anything towards solving this? And do you already know how to show the Euclidean algorithm works in simpler cases like $\mathbf Z$ and $K[x]$ ($K$ a field)? You need to understand the details of more basic cases before you try to explain harder cases.