user121926
  • Member for 8 years
  • Last seen more than 8 years ago
Is $\sqrt{2}\in{\Bbb Z}[\sqrt{2}+\sqrt{3}]$ true?
11 votes

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
Prove or disprove the implication:
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
Element in field of quotients is transcendental
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
Example on closure of a subset of a subspace of a topological space in Munkres's Topology
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
Polynomial in $\mathbb{Q}[x]$ with root $\sqrt[3]{2}+\sqrt[3]{3}$
Accepted answer
3 votes

Expand $[(x- \sqrt[3]{2})^3 - 3][(x- j\sqrt[3]{2})^3 - 3][(x- j^2\sqrt[3]{2})^3 - 3] = 0$, where $j$ is a third root of unity in $\mathbf{C}$.

View answer
Nonexistence of a vector space isomorphism
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
Area of the portion between a solid cylinder and Surface $z=x^2-y^2$
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 $...

View answer
An example of a bijective function with an infinite number of discontinuity
1 votes

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
Need help with integration
1 votes

$u = \sqrt{\frac{e^x - 1}{e^x + 1}}$ will work. Then you'll need partial fractions.

View answer
(Highschool Pre-calculus) Solving quadratic via completing the square
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
On scalar extension of module and annihilator
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
Linear operator with dense range but not full range
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
Some questions about elementary set theory (cardinal and ordinal numbers)
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
Inequality proof involving series
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
Curvature of plane curve
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
number of ways of crossing river
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
gre quanitative question confusion
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 $$1.15 \times 0.85 \times 800000 = 782000.$$

View answer