= Solution
To compute the <Ideal class group of Q of square root thirty-five>, put $\theta=\sqrt{35}$. The <ring of integers of a quadratic field> formula gives $R=\mathbb Z[\theta]$, and the <field discriminant> is $4\cdot35=140$. Both <field embeddings> are real, so the <Minkowski bound for ideal classes> is
$$
\frac{2!}{2^2}\sqrt{140}=\sqrt{35}<6.
$$
Every <ideal class> therefore has an <integral ideal> representative of <ideal norm> at most five. By <prime ideal factorization> and multiplicativity of the <ideal norm>, the <ideal class group> is generated by <prime ideals> of <ideal norm> at most five.
Apply the <Dedekind factorization theorem> to $X^2-35$. At two and five it has a repeated linear factor, giving
$$
(2)=P_2^2,\quad P_2=(2,\theta-1),\quad N(P_2)=2;
\qquad
(5)=P_5^2,\quad P_5=(5,\theta),\quad N(P_5)=5.
$$
At three, $X^2-35\equiv X^2+1$ has no root, so $(3)$ itself is a <prime ideal> of <ideal norm> nine, outside the bound. These are all the relevant <prime ideals>: a <prime ideal> of <ideal norm> four would have to lie above two with <residue degree> two, but the factorization above two has only the single factor of degree one. Consequently $[P_2]$ and $[P_5]$ generate, and each has <order of a group element> dividing two.
The <algebraic integer> $5+\theta$ has <field norm> $25-35=-10$. It belongs to both $P_2$ and $P_5$. Hence its <principal ideal> is contained in $P_2\cap P_5=P_2P_5$. Both have <ideal norm> ten, so
$$
(5+\theta)=P_2P_5.
$$
It follows that $[P_5]=[P_2]^{-1}=[P_2]$, and the <ideal class group> is generated by one element of <order of a group element> at most two.
Finally, $P_2$ is not a <principal ideal>. If $P_2=(a+b\theta)$ with $a,b\in\mathbb Z$, the equality of <ideal norm> and absolute <field norm> would imply
$$
a^2-35b^2=\pm2.
$$
Modulo five this requires $a^2\equiv2$ or $3$, whereas squares modulo five are $0,1,4$. Thus $[P_2]\ne1$, proving
$$
\boxed{\operatorname{Cl}(\mathbb Q(\sqrt{35}))\simeq\mathbb Z/2\mathbb Z}.
$$
This computes the ordinary <ideal class group>, with no positivity restriction on generators.
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