Intereting Posts

Find $f$ where $f'(x) = f(1+x)$
Understanding the intuition behind math
For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$
If the subgroup $H$ of $G$ is open in pro-$p$ topology, does it inherit the pro-$p$ topology?
Poisson Integral is equal to 1
Ring of continuous maps and prime ideals
$\lim_{{n}\to {\infty}}\frac{1^p+2^p+\cdots+n^p}{n^{p+1}}=?$
Palindromic Numbers – Pattern “inside” Prime Numbers?
is uniform convergent sequence leads to bounded function?
What is $\int_0^1 \ln (1-x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx$?
Why do odd dimensions and even dimensions behave differently?
Evaluating $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$
$(X,\tau)$ needs to be $T_1$ in order to guarantee that $A'$ is closed?
Proof for divisibility by $7$
Summation Theorem how to get formula for exponent greater than 3

I know that the set of square roots of distinct square-free integers is linearly independent over $\mathbb{Q}$. To generalize this fact, define

$R_n = \{ \sqrt[n]{s} \mid s\text{ integer with prime factorization }s = p_1^{a_1} \ldots p_k^{a_k}, \text{ where } 0 \leq a_i < n \}$

For example, $R_2$ is the set of square roots of square-free integers.

- What are zero divisors used for?
- Are free products of finite groups virtually free?
- Does $\gcd(a,bc)$ divides $\gcd(a, b)\gcd(a, c)$?
- Why is the quotient map $SL_n(\mathbb{Z})$ to $SL_n(\mathbb{Z}/p\mathbb Z)$ is surjective?
- Understanding of extension fields with Kronecker's thorem
- Galois group of $x^4-2$

Question: Is $R_n$ linearly independent over $\mathbb{Q}$ for all $n \geq 2$?

Harder (?) question: Is $\cup_{n\geq2}R_n$ linearly independent over $\mathbb{Q}$?

- Commutativity of “extension” and “taking the radical” of ideals
- Maximal sum of positive numbers
- Bijection between ideals of $R/I$ and ideals containing $I$
- Proof strategy - Stirling numbers formula
- Given $n\in \mathbb N$, is there a free module with a basis of size $m$, $\forall m\geq n$?
- diophantine equation $ |x^2-py^2|=\frac{p-1}{2} $
- Polynomial with a root modulo every prime but not in $\mathbb{Q}$.
- Prove that field of complex numbers cannot be equipped with an order relation
- GCD computations in $\mathbb{Z}$
- Understanding the ideal $IJ$ in $R$

This is true iff the radicands are multiplicatively independent. See the references to the work of Besicovitch, Mordell and Siegel in my answer to a similar question. Nowadays these results are special cases of the Galois theory of radical extensions (a.k.a. Kummer extensions).

We’ll show that incommensurable real radicals are linearly independent.

Let $\alpha_i$ real numbers, $F$ subfield of $\mathbb{R}$ so that for any $i$ $\alpha_i^{n_i}\in F$ for some $n_i>1$. Assume moreover that $\frac{\alpha_i}{\alpha_j}\not \in F$ for all $i\ne j$. Then the $\alpha_i$ are linearly independent over $F$.

For the proof we use the Lemma: let $\beta$ a real number so that $\beta^m\in F$ for some $m>1$, and $\beta\not \in F$. Then $\operatorname{Trace}_F \beta = 0$. A proof is given below.

Let now $\sum a_i \alpha_i$ a linear relation. Take $i_0 \in I$. We get

$$ a_{i_0} = \sum_{i \ne i_0} a_i \frac{\alpha_i}{\alpha_{i_0}}$$

Taking $\operatorname{Trace}^K_F$ on both sides ( $K$ is an arbitrary finite extension of $F$ containing all the $\beta_i =\frac{\alpha_i}{\alpha_{i_0}}$ we get, using lemma $d\cdot a_{i_0} = 0$, and so $a_{i_0} = 0$.

Proof of the lemma: May assume $\beta > 0$. Let $m>1$ minimal so that $\beta^m \in F$. The polynomial $X^m – \beta^m$ factors over $\mathbb{C}$ as $\prod_{j=0}^{m-1} ( X- \beta \omega^j)$. Assume that some factor $\prod_{j \in J} (X – \beta \omega^j)$ is in $F[X]$. The the free term $\prod_{ j \in J} ( – \beta \omega^j)$ is in $F$. Taking complex absolute values on both sides, we get $\beta^l \in F$ for some $1\le l < m$, contradiction. Now we know the basis $1$, $\beta$, $\ldots$, $\beta^{m-1}$ for $F(\beta)$ over $F$. It follows right away that the trace of $\beta$ is $0$.

Note: the condition $\beta$ real is necessary, as we see for $\beta = 1+i$, $\beta^4 = -4$, and $\operatorname{trace}^{\mathbb{Q}(i)}_{\mathbb{Q}} \beta = 2$.

- Why isn't there a continuously differentiable injection into a lower dimensional space?
- Does one of $L^\infty$ and $L^p, p \in (0, \infty)$ contain the other?
- Show that the derivatives of a $C^1$ function vanish a.e. on the inverse image of a null set
- Difficulties in a proof by mathematical induction (involves evaluating $\sum r3^r$).
- Show that every monotonic increasing and bounded sequence is Cauchy.
- A samurai cuts a piece of bamboo
- Bounded partial derivatives imply continuity
- Is $\ell^1$ isomorphic to $L^1$?
- Suppose $\lim \sup_{n \to \infty}a_n \le \rho$. Show $\lim \sup_{n \to \infty} a_n^{{(n-m)}/{n}} \le \rho$.
- Derivation of binomial coefficient in binomial theorem.
- Calculate an integral with Riemann sum
- $f(x)=1/(1+x^2)$. Lagrange polynomials do not always converge. why?
- Difference Between Limit Point and Accumulation Point?
- I want to calculate the limit of: $\lim_{x \to 0} \left(\frac{2^x+8^x}{2} \right)^\frac{1}{x} $
- $f(x)=3x+4$ – Injective and Surjective?