Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q  \implies  \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $ Assume that $a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q$ are rational，prove $\sqrt{a},\sqrt{b},\sqrt{c}  \in\mathbb Q$,are rational.
I know that can be proved, would like to know that there is no easier way

$\sqrt a + \sqrt b + \sqrt c = p \in \mathbb Q$,
$\sqrt a + \sqrt b  = p- \sqrt c$,
$a+b+2\sqrt a \sqrt b = p^2+c-2p\sqrt c$,
$2\sqrt a\sqrt b=p^2+c-a-b-2p\sqrt c$,
$4ab=(p^2+c-a-b)+4p^2c-4p(p^2+c-a-b)\sqrt c$,
$\sqrt c=\frac{(p^2+c-a-b)+4p^c-4ab}{4p(p^2+c-a-b)}\in\mathbb Q$.
 A: [See  here and here for an introduction to the proof. They are explicitly worked special cases]
As you surmised, induction works, employing our prior Lemma (case $\rm\:n = 2\:\!).\:$ Put $\rm\:K = \mathbb Q\:$ in
Theorem $\rm\ \sqrt{c_1}+\cdots+\!\sqrt{c_{n}} = k\in K\ \Rightarrow \sqrt{c_i}\in K\:$ for all $\rm i,\:$ if $\rm\: 0 < c_i\in K\:$  an ordered field.
Proof $\: $ By induction on $\rm n.$ Clear if $\rm\:n=1.$ It is true for $\rm\:n=2\:$ by said Lemma. Suppose that $\rm\: n>2.$ It suffices to show one of the square-roots is in $\rm K,\:$ since then the sum of all of the others is in $\rm K,\:$ so, by induction, all of the others are in $\rm K$.
Note that $\rm\:\sqrt{c_1}+\cdots+\sqrt{c_{n-1}}\: =\: k\! -\! \sqrt{c_n}\in K(\sqrt{c_n})\:$ so all $\,\rm\sqrt{c_i}\in K(\sqrt{c_n})\:$ by induction.
Therefore $\rm\ \sqrt{c_i} =\: a_i + b_i\sqrt{c_n}\:$ for some $\rm\:a_i,\:\!b_i\in K,\:$ for $\rm\:i=1,\ldots,n\!-\!1$.
Some $\rm\: b_i < 0\:$ $\Rightarrow$ $\rm\: a_i = \sqrt{c_i}-b_i\sqrt{c_n} = \sqrt{c_i}+\!\sqrt{b_i^2 c_n}\in K\:\Rightarrow \sqrt{c_i}\in K\:$ by Lemma $\rm(n=2).$
Else all $\rm b_i \ge 0.\:$ Let $\rm\:  b = b_1\!+\cdots+b_{n-1} \ge 0,\:$ and let $\rm\: a = a_1\!+\cdots+a_{n-1}.\:$ Then
$$\rm \sqrt{c_1}+\cdots+\!\sqrt{c_{n}}\: =\: a+(b\!+\!1)\:\sqrt{c_n} = k\in K\:\Rightarrow\:\!\sqrt{c_n}= (k\!-\!a)/(b\!+\!1)\in K$$
Note $\rm\:b\ge0\:\Rightarrow b\!+\!1\ne 0.\:$ Hence, in either case, one of the square-roots is in $\rm K.\ \ $ QED
Remark $ $ Note that the proof depends crucially on the positivity of the square-root summands. Without such the proof fails, e.g. $\:\sqrt{2} + (-\sqrt{2})\in \mathbb Q\:$ but $\rm\:\sqrt{2}\not\in\mathbb Q.\:$ It is instructive to examine all of the spots where positivity is used in the proof (above and Lemma), e.g. to avoid dividing by $\,0$.
See also this post on linear independence of square-roots (Besicovic's theorem).
A: Maybe not easier, but quite elegant :
Suppose that $a,b,c$ are all non zero. Let $K=\mathbb{Q}(\sqrt{a},\sqrt{b},\sqrt{c})$ and $n = [K: \mathbb{Q}]$. Then since $Tr_{K/\mathbb{Q}}(\sqrt{a}) = Tr_{\mathbb{Q}(\sqrt{a})/\mathbb{Q}} \circ Tr_{K/\mathbb{Q}(\sqrt{a})} (\sqrt{a})$, we have
$$ Tr_{K/\mathbb{Q}}(\sqrt{a}) = \begin{cases} 0,& \text{if } \sqrt{a} \notin \mathbb{Q} \\ n\sqrt{a}, &\text{if } \sqrt{a} \in \mathbb{Q}, \end{cases}$$
and same for $\sqrt{b}$ and $\sqrt{c}$.
By hypothesis $\sqrt{a} + \sqrt{b} +\sqrt{c} \in \mathbb{Q}$, so
$$ Tr_{K/\mathbb{Q}}(\sqrt{a}) + Tr_{K/\mathbb{Q}}(\sqrt{b}) + Tr_{K/\mathbb{Q}}(\sqrt{c}) = n\sqrt{a} + n \sqrt{b} + n \sqrt{c}.$$
It is easy to conclude that $\sqrt{a},\sqrt{b},\sqrt{c} \in \mathbb{Q}$.
