Linked Questions

1
vote
2answers
708 views

An element not in a field extension [duplicate]

Possible Duplicate: Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$? Consider the field extension $\mathbb{Q}(\sqrt2)$. I want to show that $\sqrt5 \notin \mathbb{Q}(\sqrt2)$. If this were ...
-1
votes
1answer
196 views

$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$ change a sign to be rational [closed]

I have this problem: $$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$$ Prove that you need to change ONLY a sign (to convert a $+$ to $-$) of a single square root, for the sum to be rational. EDIT:...
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 ...
13
votes
2answers
257 views

Proving that if $x_1,\dots,x_n$ are rational numbers and $\sqrt{x_1}+\dots\sqrt{x_n}$ is rational, then each $\sqrt{x_i}$ is rational as well

I'm having a hard time with the following problem: Let $x_1,x_2...x_n$ be rational numbers. Prove that if the sum $\sqrt{x_1}+\sqrt{x_2}+...+\sqrt{x_n}$ is rational, then all $\sqrt{x_i}$ are ...
7
votes
1answer
531 views

Is the sum of the square roots of all natural numbers up to n whole for any value of n other than 1?

For the summation $\sum_{n=0}^x \sqrt{n}$ are there any values of $x$ where the summation equals a whole number other than 1?
3
votes
5answers
166 views

show that $\sqrt{3} \notin \mathbb{Q}(\sqrt{2})$

I'd like to prove that $\sqrt{3}$ is not in the field $\mathbb{Q}(\sqrt{2})$. Let's write out a system of equations: $$ x^2 = 3 y^2 \text{ with } x,y \in \mathbb{Z}[\sqrt{2}] $$ Then if we write $x =...
3
votes
1answer
129 views

Show that $\sqrt{3}$ is not an element of the $\text{span} (1, \sqrt{2})$

So we have the following given to us this weekend just on a handout: Considering $\mathbb R$ as a vector space over the field $\mathbb Q$, show that $\sqrt 3$ is not an element of the span of $(1, \...
4
votes
1answer
246 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 $\mathbb Q$?
1
vote
1answer
235 views

How to get this equation solved?

I came across this equation $$\sqrt a-\sqrt b=\sqrt 7-\sqrt 5$$ And you have to find the value of '$a$' and '$b$' when both of them are primes. The solution was $a=7, b=5$. Now, my question is, ...
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 $i \notin \mathbb{Q}(i\sqrt{7})$?
9
votes
2answers
96 views

Are square root binomials unique?

In Euclid we find the notion of a binomial, its simply a sum $s = \sqrt{a}+\sqrt{b}$ of two square roots $\sqrt{a}$ and $\sqrt{b}$. Lets say such a sum is simple iff $a$ and $b$ are positive non-zero ...
6
votes
1answer
604 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 $...
3
votes
2answers
270 views

Algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. Alternative proof?

Let $\mathcal{A}$ be the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. Prove that $[\mathcal{A}:\mathbb{Q}] = \infty$. I can show this using $[\mathbb{Q}(\sqrt[n]{2}):\mathbb{Q}] = n$ for all $n ...
3
votes
3answers
127 views

$[E:\mathbb{Q}]$, the degree of $E$ over $\mathbb{Q}$

Algebraic Extension $E=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$, I need to find the $[E:\mathbb{Q}]$, the degree of $E$ over $\mathbb{Q}$ I know degree of $\mathbb{Q}(\sqrt{2})$ over $\mathbb{Q}$ is $...
2
votes
1answer
502 views

sum of square root of primes 2

I dont know how to solve the problem below. (1) $p[1]$, $p[2]$, $\ldots$, $p[n]$ are distinct primes, where $n = 1,2,\ldots$ Let $a[n]$ be the sum of square root of those primes, that is, $a[n] = \...

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