Letting $\alpha = \sqrt{2} + \sqrt{3}$, we have $\sqrt{2} = \frac{1}{2}\alpha^3 - \frac{9}{2}\alpha \in \mathbf{Q}(\alpha)$. The minimal polynomial of $\alpha$ over $\mathbf{Q}$ is $f(x) = x^4 - 10x^... View answer 5 votes I will follow the method suggested by Blue in his comment to show that the implication does not hold. The sides$a$,$b$,$c$are proportional to$\sin A$,$\sin B$and$\sin C$, respectively, by the ... View answer 3 votes Write$x = \frac{p(c)}{q(c)}$, where$p$and$q$are relatively prime polynomials. Then we get $$d_0 q^k + d_1 p q^{k-1} + \cdots + d_k p^k = 0, \quad k \geq 1, \quad d_0, d_k \ne 0.$$ Then$q | p^k$... View answer Accepted answer 3 votes$0$does belong to$\overline{A}$where this denotes the closure of$A$in$\mathbf{R}$. However, if by$\overline{A}$we mean the closure of$A$in$Y$, then$0 \not\in \overline{A}$. The theorem you ... View answer Accepted answer 3 votes Expand$[(x- \sqrt{2})^3 - 3][(x- j\sqrt{2})^3 - 3][(x- j^2\sqrt{2})^3 - 3] = 0$, where$j$is a third root of unity in$\mathbf{C}$. View answer Accepted answer 3 votes The two vector spaces are certainly isomorphic (though I haven't constructed an isomorphism). If a$\mathbf{Q}$-vector space$V$has an infinite basis$I$, then$V$and$I$have same cardinality. ($V$... View answer Accepted answer 2 votes You've chosen to parametrize using polar coordinates, which is a good idea. The solid cylinder$x^2 + y^2 \leq 1$extends above the disk$D$defined by$x^2 + y^2 \leq 1$, which is located in the$...
For the second part, it is easy to find a bijection between $\mathbf{R}$ and $(0,+\infty)$. (The exponential map is an example.) So you just need to find a bijection between $[0,+\infty)$ and $(0,+\... View answer 1 votes$u = \sqrt{\frac{e^x - 1}{e^x + 1}}$will work. Then you'll need partial fractions. View answer 1 votes$x^2 - 6x$isn't the same as$(x-3)^2$, so your equation$(x-3)^2 = 16$is wrong.$(x-3)^2$is actually$x^2 - 6x + 9$, so you should write $$x^2 - 6x + 9 = 25$$ and then $$(x-3)^2 = 25$$. So your ... View answer Accepted answer 1 votes No, this is false. Consider the element$1 \otimes 2 \in \mathbf{Z}/2\mathbf{Z} \otimes_{\mathbf{Z}} \mathbf{Z}$. (We have$1 \otimes 2 = 2 \otimes 1 = 0 \otimes 1 = 0$, but the only element in the ... View answer 0 votes Let$E$be the space of continuous functions$f$on$[0,1]$satisfying$f(0) = 0$, with the sup norm. Now define$A(f)(x) = \int_{0}^x f(t) \, dt$. View answer Accepted answer 0 votes I will only answer question 1. The answer is yes. Either$X$or its complement must have cardinality$2^{\aleph_0}$. For example, say it's$X$. Then the order type of$X$is at least$2^{\aleph_0}$, ... View answer 0 votes Hint. First note that the required bound is independent of$n$. So you won't need to use$n$in an essential way. In the worst case,$n$will be very large, so you might as well write down the ... View answer 0 votes The length$\sqrt{(\gamma_1(t+\Delta t)-\gamma_1(t))^2+(\gamma_2(t+\Delta t)-\gamma_2(t))^2}$is the distance between the point$\gamma(t + \Delta t)$and$\gamma(t)$. This is not the distance they're ... View answer Accepted answer 0 votes He needs to take six steps, one or two at a time. The possibilities are: $$2 + 2 + 2 = 6 \quad \text{(one possibility)}$$ $$2 + 2 + 1 + 1 = 6 \quad \text{(six possibilities, depending on the order)... View answer 0 votes To decrease a number by 15\%, multiply it by 0.85. To increase it by 15\%, multiply it by 1.15. So the answer is$$1.15 \times 0.85 \times 800000 = 782000.$\$