Linked Questions

6
votes
1answer
5k views

Showing field extension $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})/\mathbb{Q}$ degree 8 [duplicate]

Possible Duplicate: The square roots of the primes are linearly independent over the field of rationals I am trying to classify the Galois group of the field extension $\mathbb{Q}(\sqrt{2}, \sqrt{...
3
votes
3answers
270 views

Why is $n_1 \sqrt{2} +n_2 \sqrt{3} + n_3 \sqrt{5} + n_4 \sqrt{7} $ never zero? [duplicate]

Here $n_i$ are integral numbers, and not all of them are zero. It is natural to conjecture that similar statement holds for even more prime numbers. Namely, $$ n_1 \sqrt{2} +n_2 \sqrt{3} + n_3 \...
4
votes
1answer
319 views

Is there a way to show the sum of any different square root of prime numbers is irrational? [duplicate]

Is there a way to show the sum of any different square root of prime numbers is irrational? For example, $$\sqrt2+\sqrt3+\sqrt5 +\sqrt7+\sqrt{11}+\sqrt{13}+\sqrt{17}+\sqrt{19}$$ should be a irrational ...
0
votes
1answer
370 views

root of prime numbers are linearly independent over $\mathbb{Q}$ [duplicate]

How can we prove by mathematical induction that $1,\sqrt{2}, \sqrt{3}, \sqrt{5},\ldots, \sqrt{p_n}$ ($p_n$ is the $n^{\rm th}$ prime number) are linearly independent over the rational numbers ? $\...
2
votes
1answer
302 views

$\sqrt{p_1}$ is not in $Q[\sqrt{p_2},…,\sqrt{p_n}]$ [duplicate]

How to show $\sqrt{p_1}$ is not in $Q[\sqrt{p_2},...,\sqrt{p_n}]$ if $p_1,...,p_n$ are distinct primes? Intuitively, this is pretty clear, but it makes me very uncomfortable to just believe. Any idea ...
2
votes
1answer
143 views

Exercise about field extensions [duplicate]

Consider $a_1,\ldots,a_n\in \mathbb Z$. i) Suppose $a_1,\ldots, a_n$ are pairwise relatively prime. I have to see by induction on n that $[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q]=2^n$ Once ...
1
vote
0answers
86 views

Linear independence of square root of square free numbers [duplicate]

Possible Duplicate: The square roots of the primes are linearly independent over the field of rationals I am reading a research article in which there is a theorem regarding square roots of ...
3
votes
0answers
73 views

Linear independence in $\mathbb{R}$ [duplicate]

Possible Duplicate: The square roots of the primes are linearly independent over the field of rationals I would like to prove that the family $\{\sqrt{p}, p\text{ prime number} \}$ is linearly ...
1
vote
0answers
44 views

If $k_1, \ldots, k_n$ are non-square, pairwise coprime, then $\sqrt {k_n} \not \in \mathbf{Q}(\sqrt {k_1}, \ldots, \sqrt {k_{n-1}})$ [duplicate]

Seems intuitive. Like the fact that $\sqrt 3 \not \in \mathbf{Q}(\sqrt 2)$. But how to approach actually proving it? The proof of this fact doesn't seem to generalize well.
124
votes
6answers
28k views

Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?

It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?
21
votes
6answers
4k views

Is there a way to write an infinite set that contains only irrational numbers without integer multiples?

Is there a way to write an infinite set that contains only irrational numbers without integer multiples? The infinite set must not contain integer multiples of any other members of that set. For ...
26
votes
5answers
5k views

Can a finite sum of square roots be an integer?

Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer? The square roots need to be irrational.
19
votes
3answers
5k views

How to prove $1$,$\sqrt{2},\sqrt{3}$ and $\sqrt{6}$ are linearly independent over $\mathbb{Q}$?

How do I prove that $1$,$\sqrt{2},\sqrt{3}$ and $\sqrt{6}$ are linearly independent over $\mathbb{Q}$? $\mathbb{Q}$ is the rational field. I want to know the detail about the proof. Thanks in advance....
27
votes
3answers
2k views

Is it possible for integer square roots to add up to another?

I initially was wondering if it were possible for there to be three $x,y,z \in \mathbb{Q}$ and $\sqrt{x},\sqrt{y},\sqrt{z} \notin \mathbb{Q}$ such that $\sqrt{x} + \sqrt{y} = \sqrt{z}$. I had ...
40
votes
4answers
4k views

Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.

I have read the following theorem: If $p_1,p_2,\dots,p_n$ are distinct prime numbers, then$$\left(\mathbb Q\left[\sqrt p_1,\dots,\sqrt p_n\right]:\mathbb Q\right)=2^n.$$ I have tried to prove a ...

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