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 Jul12 asked Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$ Jul11 comment Help proving that $\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{1}{t}\cos(\frac{\pi}{t}))^2}dt$ Ya you're right, I completely @#$%ed this whole problem up Jul11 comment Help proving that$\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{1}{t}\cos(\frac{\pi}{t}))^2}dt$Shoot well you're right I don't know what is going on then, this integral is the arc length for$(t,t\sin(\frac{\pi}{t}))$and according to the book (Do Carmo's Differential Geometry), the inequality should hold. so yah I don't know then Jul11 comment Help proving that$\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{1}{t}\cos(\frac{\pi}{t}))^2}dt$I'm just getting a series expansion, how do you get wolfram definite integrator to approximate it? Jul11 comment Help proving that$\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{1}{t}\cos(\frac{\pi}{t}))^2}dt$Ok actually it is for$n\geq 1$so I'm not sure what to think then, what did you use to check the integral? Jul11 revised Help proving that$\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{1}{t}\cos(\frac{\pi}{t}))^2}dt$added 45 characters in body Jul11 revised Help proving that$\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{1}{t}\cos(\frac{\pi}{t}))^2}dt$deleted 12 characters in body Jul11 awarded Teacher Jul11 revised Help proving that$\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{1}{t}\cos(\frac{\pi}{t}))^2}dt$added 210 characters in body Jul11 revised Help proving that$\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{1}{t}\cos(\frac{\pi}{t}))^2}dt$added 14 characters in body Jul11 asked Help proving that$\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{1}{t}\cos(\frac{\pi}{t}))^2}dt$Jul7 comment If$E$and$F$are subfields of a finite field$K$and$E\cong F$, prove that$E = F$Oh that's right, a splitting field is the smallest field possible, ok cool thanks. Your conciseness is legendary André, unfortunately us mere mortals sometimes require a bit of redundancy in the answers we receive. (said in jest =]) Jul7 accepted If$E$and$F$are subfields of a finite field$K$and$E\cong F$, prove that$E = F$Jul7 comment If$E$and$F$are subfields of a finite field$K$and$E\cong F$, prove that$E = F$That result is one of my theorems yes, and Zev's comment answers my question (thank you Zev). But is knowing both contain all the roots of that poly enough without knowing it's separable? Not that it's particularly important, anyways thanks for the help. Jul7 comment If$E$and$F$are subfields of a finite field$K$and$E\cong F$, prove that$E = F$It seems intuitively obvious, but how do we know all the roots of$x^q - x$are distinct? Jul7 revised If$E$and$F$are subfields of a finite field$K$and$E\cong F$, prove that$E = F$added 42 characters in body Jul7 comment If$E$and$F$are subfields of a finite field$K$and$E\cong F$, prove that$E = F$Sorry I should have mentioned, I have no Galois Theory at my disposal. Group Theory up through Sylow Theorems, and Field Theory through splitting fields and general results on the classification of finite fields. Also undergrad ring theory, but that's probably not important. Jul7 asked If$E$and$F$are subfields of a finite field$K$and$E\cong F$, prove that$E = F$Jul6 accepted A property equivalent to$K$being a finite-dimensional, normal extension field of$F$Jul4 accepted Finding a splitting field of$x^3 + x +1$over$\mathbb{Z}_2\$