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{...
6
votes
1answer
501 views

Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers. [duplicate]

Take $p_1, p_2, \ldots, p_n, p_{n+1}$ be $n+1$ prime numbers in $\mathbb{P} \subseteq \mathbb{N}$. $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ seems to be quite ...
6
votes
1answer
97 views

Sum of 5 square roots equals another square root. What is the minimum possible value of the summed square root?

In the equation $\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}+\sqrt{e}=\sqrt{f}$, each variable is a distinct positive integer. What is the least possible value for $f?$ Out of purely trial and error, I have ...
6
votes
1answer
614 views

Elementary proof that finite sums of square roots of primes is irrational

It is relatively easy to show that if $p_1$, $p_2$ and $p_3$ are distinct primes then $\sqrt{p_1}+\sqrt{p_2}$ and $\sqrt{p_1}+\sqrt{p_2}+\sqrt{p_3}$ are irrational, but the only proof I can find that $...
5
votes
7answers
665 views

Proving that $\sqrt[3] {2} ,\sqrt[3] {4},1$ are linearly independent over rationals

I was trying to prove that $\sqrt[3] {2} ,\sqrt[3] {4}$ and $1$ are linearly independent using elementary knowledge of rational numbers. I also saw this which was in a way close to the question I was ...
5
votes
6answers
655 views

Does $\mathbb Q(\sqrt{-2})$ contain a square root of $-1$?

This isn't a homework question but one I found online. Does $\mathbb Q(\sqrt{-2})$ contain a square root of $-1$? We just started doing field theory in my class and I want extra practice, but I ...
4
votes
3answers
858 views

The sum of square roots of non-perfect squares is never integer [duplicate]

My question looks quite obvious, but I'm looking for a strict proof for this: Why can't the sum of two square roots of non-perfect squares be an integer? For example: $\sqrt8+\sqrt{15}$ isn't an ...
4
votes
2answers
1k views

Linear independence of roots over Q

Let $p_1,\ldots,p_k$ be $k$ distinct primes (in $\mathbb{N}$) and $n>1$. Is it true that $[\mathbb{Q}(\sqrt[n]{p_1},\ldots,\sqrt[n]{p_k}):\mathbb{Q}]=n^k$? (all the roots are in $\mathbb{R}^+$) ...
4
votes
5answers
114 views

Let $a_0+a_1x+…+a_nx^n$ be a non zero polynomial with integer coefficients.if $p(√2+√3+√6)=0$, the smallest possible value of n is?

Question Let $a_0+a_1x+....+a_nx^n$ be a non zero polynomial with integer coefficients.if $p(√2+√3+√6)=0$, the smallest possible value of n is? Honestly I have no idea how to begin to solve this ...
4
votes
4answers
84 views

Are these two fields the same?

I wanted to know if the field $\mathbb{Q}(i\sqrt{7}) = \mathbb{Q}(\sqrt{7}, i)$ are the same. I don't think they are because $i \notin \mathbb{Q}(i\sqrt{7})$?
4
votes
1answer
247 views

Infinitely many transcendental numbers over Q

My previous question was not well-framed so I will ask again: Can you explicitly produce an infinite set of real numbers which is algebraically independent over $\mathbb Q$?
4
votes
1answer
2k views

Finding a primitive element for the field extension $\mathbb{Q}(\sqrt{p_{1}},\sqrt{p_{2}},\ldots,\sqrt{p_{n}})/\mathbb{Q}$

Let $p_1,\ldots,p_n\in\mathbb{N}$ be different prime numbers, it can be shown that $[\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_n}):\mathbb{Q}]=2^n$ and in any case it is clearly finite since $[\mathbb{Q}(\...
4
votes
1answer
365 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 ...
4
votes
1answer
188 views

$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational

If $m_1 , m_2, \cdots m_n$ are natural numbers where at least one of them is not a perfect square, then how do I prove that the sum $$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$$ is irrational? I'm ...
3
votes
3answers
4k views

Finding a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$.

I have to find a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$. I determined that $\sqrt{2}+\sqrt{3}$ satisfies the equation $(x^2-5)^2-24$ in $\Bbb{Q}$. Hence, the basis should be $1,(\...

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