Can the sum of two irrational roots of two coprime natural numbers greater than 1 be an irrational root of a rational number?

Question

After thinking more about my last question and reading the answers I reformulated it. I'm positive I wanted to ask the following:

Let $x,y\in \mathbb{N};\: x,y>1;\: gcd(x,y)=1; \: \sqrt{x},\sqrt{y}\notin\mathbb{N}$ and let $z\in\mathbb{Q}; \: \sqrt{z}\notin\mathbb{Q}$.
Is it possible to find $x, y, z$ obeying the rules above such that the following equality is true?

$\sqrt{x}+\sqrt{y}=\sqrt{z}$

If so, find a sufficient and necessary condition in the choosing of $x$ and $y$ for $z$ to exist.

Basically I want to know if we can find $r$ rational which makes the equality true in expressions like $\sqrt{2}+\sqrt{3}=\sqrt{r} \: and \: \sqrt{5}+\sqrt{11}=\sqrt{r}$.

Answer 1

In addition to the elementary solution mentioned in the comments (square both sides and find an irrational number on one side and a rational on the other), you might be interested in more of the theory behind this sort of question. The short version is that any polynomial equation $P(x)=0$ with integer coefficients satisfied by a term like $\sqrt{2}+\sqrt{3}$ must have a polynomial of degree at least four, whereas any number of the form $\sqrt{r}$ with $r$ rational satisfies an equation $P(x)=0$ with $P()$ of degree two. The degree of the smallest polynomial that an algebraic number is a root of is also known as the degree of the algebraic number, and searching on 'algebraic number degree' should find you more good information on this sort of thing.

Incidentally, on a somewhat-interesting related note, while it's possible to determine which side of $\sqrt{r}$ such an expression lies on — or more generally, whether $a\sqrt{x}+b\sqrt{y}+\ldots$ is greater than or less than zero for arbitrary rationals $a, b, \ldots$ and $x, y, \ldots$ — it's also a 'hard' problem in the sense that nobody knows an efficient algorithm to do it, or even how efficient such an algorithm can be! The general phrase for this question is the Sum of Square Roots problem; see e.g. http://cs.smith.edu/~jorourke/TOPP/P33.html or https://cstheory.stackexchange.com/questions/4053/sum-of-square-roots-hard-problems .

Answer 2

Hint:

If $\displaystyle \sqrt{x}+\sqrt{y}=\sqrt{z}$ then squaring both sides gives $$x+y+2\sqrt{xy}=z$$

Note that $x+y\in \Bbb{N}$ and $z\in \Bbb{Q}$. So if $\gcd(x,y)=1$ then $\sqrt{xy}\in\Bbb{R} \setminus\Bbb{Q}$

So what do we infer from this?