The definition of "natural" is of course somewhat personal and involves perhaps aesthetic aspects and/or interesting properties. I illustrate it with two examples:

The integer $((p-1)/2)!$ is such a square-root of $-1$ (modulo $p$ for $p$ a prime congruent to $1$ modulo $4$).

Another choice which is perhaps natural is given by $2a/b\pmod p$ where $p=4a^2+b^2$ with $a,b\in\mathbb N$.

Are there other nice formulae?

An example of a not very natural choice (in my eyes) is $a^{(p-1)/4}\pmod p$ where $a$ is the smallest natural integer generating the multiplicative group of invertible elements modulo $p$. However, I accept this answer gladly, if there is some generator $a$ of $(\mathbb Z/p\mathbb Z)^*$ given by a "natural" formula.

(The motivation for this question was Does (the ideal class of) the different of a number field have a canonical square root? )

The definition of "natural" is of course somewhat personal and involves perhaps aesthetic aspects and/or interesting properties. I illustrate it with two examples:

The integer $((p-1)/2)!$ is such a square-root of $-1$ (modulo $p$ for $p$ a prime congruent to $1$ modulo $4$).

Another choice which is perhaps natural is given by $2a/b\pmod p$ where $p=4a^2+b^2$ with $a,b\in\mathbb N$.

Are there other nice formulae?

An example of a not very natural choice (in my eyes) is $a^{(p-1)/4}\pmod p$ where $a$ is the smallest natural integer generating the multiplicative group of invertible elements modulo $p$. However, I accept this answer gladly, if there is some generator $a$ of $(\mathbb Z/p\mathbb Z)^*$ given by a "natural" formula.

(The motivation for this question was Does (the ideal class of) the different of a number field have a canonical square root? )