Euclidean numbers are those real number that can be constructed from the natural numbers by a finite chain of +,-,*,/ and $\sqrt{}$. They are also called Constructible Numbers.
I am now interested in whether there is an (easy) algorithm to determine if two Euclidean Numbers are equal (as real numbers) or, equivalently, if a given Euclidean Number is zero.
If $\alpha$ is constructible, you can construct a minimal polynomial of $\alpha$ recursively as follows: If $\alpha=\sqrt{\beta}$, and $\beta$ has minimal polynomial $P(x)$, then $P(x^2)$ is a polynomial with $P(\alpha)=0$. If $\alpha=\beta+\gamma$, and $\beta, \gamma$ have minimal polynomials $P, Q$ with roots $x_1, \ldots, x_n$ and $y_1, \ldots, y_m$, respectively, then $R(x) = \prod_{i,j}(x-x_i-y_j)$ is a polynomial with $R(\alpha)=0$. Note that using elementary symmetric functions you can compute $R$ without knowing the $x_i, y_j$'s. Factorize the resulting polynomial over $\mathbb{Q}$, and check numerically which factor contains $\alpha$.
The last step uses that roots of integral polynomial cannot be too close together. An efficient version of this statement is Liouville's theorem, using this you know that you have to compute $\alpha$ only to a certain predictable precision.
Note that you cannot do without some numerical computation, because the usual convention that the square root is a map $\sqrt{\phantom{x}}:[0, \infty)\rightarrow[0, \infty)$ is quite artificial from an algebraic point of view, so distinguishing $2+\sqrt{2}$ from $2-\sqrt{2}$ requires some non-field theoretic input, like positivity or order.
This algorithm is not polynomial, as the degree of the minimal polynomial grows too fast, however, in practice it can be used for rather difficult looking expressions.
