Intereting Posts

Everywhere Super Dense Subset of $\mathbb{R}$
Check Points are line, triangle, circle or rectangle
Find the exact value of the infinite sum $\sum_{n=1}^\infty \big\{\mathrm{e}-\big(1+\frac1n\big)^{n}\big\}$
Big-O: If $f(n)=O(g(n))$, prove $2^{f(n)}=O(2^{(g(n)})$
An informal description of forcing.
Cancellation in topological product
Which finite groups are the group of units of some ring?
Prove that $\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$
Is the Structure Group of a Fibre Bundle Well-Defined?
Convergence of $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$
Is it possible to place 26 points inside a rectangle that is 20 cm by 15 cm so that the distance between every pair of points is greater than 5 cm?
Can infinitely many primes lie over a prime?
Pure algebra: Show that this expression is positive
$A$ closed subset of a metric space $(M,d)$ , let $r>0$ , then is $X(A,r):=\{x\in M : \exists a\in A$ such that $d(x,a)=r\}$ closed in $M$?
Why is the difference of distinct roots of irreducible $f(x)\in\mathbb{Q}$ never rational?

I have no idea how to do this.

To find the minimal polynomial of say $\sqrt2 + \sqrt3$, we need to find the monic polynomial $p \in \mathbb Q$ (correct if I am wrong but monic polynomial is when the coefficient of the highest degree term is $1$) of the smallest possible degree such that $\sqrt2 + \sqrt3$ is a root of $p$.

If we let $u=\sqrt2 + \sqrt3$ then $u ^2 = 5+ 2 \sqrt6 \iff u^2 – 5 = 2 \sqrt6 $, then $(u^2 – 5)^2=24 \iff u^4 -10u^2 +1=0$

- Computing the number of nonisomorphic finite abelian groups of order $n$
- $\mathbb{Z} \times \mathbb{Z}$ is cyclic.
- Left inverse implies right inverse in a finite ring
- Homogeneous polynomial in $k$ can factor into linear polynomials?
- Show that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$
- How do I show that two groups are not isomorphic?

All I did was keep squaring until all of the irrational terms go away. But what next? Am I doing this correctly and what do we do next if I am?

- Prove that $G = \langle x,y\ |\ x^2=y^2 \rangle $ is torsion free.
- Prime elements in $\mathbb{Z}$
- Show that any cyclic group of even order has exactly one element of order $2$
- Why does $K \leadsto K(X)$ preserve the degree of field extensions?
- Does $$ divide $\phi(n)$?
- Find all subrings of $\mathbb{Z}^2$
- When is a product of two ideals strictly included in their intersection?
- When are two semidirect products isomorphic?
- Legendre symbol $(-21/p)$
- On the Definition of multiplication in an abelian group

First, show that $\sqrt{3}$ is not in the quadratic extension generated by $\sqrt{2}.$ That means that the degree of the extension is at least $4.$ But you have found a polynomial of degree $4,$ so it must be minimal.

Suppose $\;\sqrt2\in\Bbb Q(\sqrt3)\;$ , then there exist $\;a,b\in\Bbb Q\;$ such that

$$\sqrt2=a+b\sqrt3\implies 2=a^2+3b^2+2ab\sqrt3\implies\sqrt3\in\Bbb Q\;,\;\;\text{contradiction}$$

Thus, $\;x^2-2\;$ must be irreducible in $\;\Bbb Q(\sqrt3)[x]\;$ , so that $\;\sqrt2+\sqrt3\;$ must belong to an extension of $\;\Bbb Q\;$ of at least degree $\;4\;$ . Since you already found a rational polynomial of degree four which vanishes at $\;\sqrt2+\sqrt3\;$ you finished, as then this must be an irreducible polynomial (otherwise this sum of square roots would belong to an extension of degree less than four).

You know that the minimal polynomial for $\sqrt3$ over $\Bbb Q$ is $X^2-3$, and we’ll believe that this is still the minimal polynomial for $\sqrt3$ over $\Bbb Q(\sqrt2\,)$. This means that the polynomial for $\sqrt3+\sqrt2$ over $\Bbb Q(\sqrt2\,)$ is $(X-\sqrt2\,)^2-3$. Expand this out, and multiply it by its “conjugate” (replacing $\sqrt2$ by $-\sqrt2\,$) and get a $\Bbb Q$-polynomial. That’s it.

- Stopping times and $\sigma$-algebras
- Free idempotent semigroup with 3 generators
- $\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$ is never zero when restricted to $\mathbb{S^2}$
- Prove $\frac{|a+b|}{1+|a+b|}<\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$.
- Why is Gödel's Second Incompleteness Theorem important?
- $f(x)=x^3+ax^2+bx+c$ has roots $a,b$ and $c$
- On the series $\sum \limits_{n=2}^{\infty} \frac{(-1)^n}{n \log n}$
- Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares
- A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous
- Rigorous proof that surjectivity implies injectivity for finite sets
- What is the vector form of Taylor's Theorem?
- Pythagorean Theorem Proof Without Words (request for words)
- Continuous bijection from $(0,1)$ to $$
- number of ways you can partition a string into substrings of certain length
- Lines tangent to two circles