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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 $\mathbb{Q}(\sqrt{2}, \sqrt{... 3answers 270 views ### Why is$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, $$n_1 \sqrt{2} +n_2 \sqrt{3} + n_3 \... 1answer 319 views ### Is there a way to show the sum of any different square root of prime numbers is irrational? [duplicate] Is there a way to show the sum of any different square root of prime numbers is irrational? For example,$$\sqrt2+\sqrt3+\sqrt5 +\sqrt7+\sqrt{11}+\sqrt{13}+\sqrt{17}+\sqrt{19}$$should be a irrational ... 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 ? \... 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 ... 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 ... 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 ... 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 ... 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. 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? 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 ... 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. 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.... 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 ... 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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