moving i's outside of surds in radical expressions

Question

Given an algebraic number expressed in radicals that has an i inside a surd, does an alternative representation of the same value that is still expressed in algebraic terms but with the i not inside a surd always exist? For example: $$\sqrt{1+i}=\sqrt[4]{2}\sqrt{2+\sqrt{2}}/2+i\sqrt[4]{2}\sqrt{2-\sqrt{2}}/2$$ If such a representation does always exist, is there a general method for finding it?

Edit: I am interested in cases where the surd is an arbitrary nth root, not just a square root as in the example.

Answer 1

If $z=\sqrt[n]w$ then, in general you are trying to solve $z^n=w=\rho(\cos\phi + i\sin\phi)$ say, i.e. express $w$ in polar form.

The $n$ roots are then

$$ z_k=\rho^\frac{1}{n} \left( \cos\left(\frac{\phi}{n} + k\frac{2\pi}{n}\right) +i\sin\left(\frac{\phi}{n} + k\frac{2\pi}{n}\right) \right)\\ k=0,\ldots,n-1 $$ with the principal root being given by $k=0$.

In your case $w=1+i = \sqrt2(\cos\pi/4+i\sin\pi/4)$

Answer 2

For $z = a + bi$, $$\sqrt[n]{z} \\= (|z|e^{i * atan2(a, b)})^{1 / n} \\= |z|^{1 / n} * e^{i * atan2(a, b) / n} \\= |z|^{1 / n} * (\cos(\frac{atan2(a, b)}{n}) + i\sin(\frac{atan2(a, b)}{n})$$ The question is equivalent to whether $\cos(\frac{atan2(a, b)}{n})$ and $\sin(\frac{atan2(a, b)}{n})$ are expressible in real radicals. In cases where $\frac{atan2(a,b)}{n}$ is a rational multiple of $\pi$, this is true if and only if the denominator of the reduced form of the rational multiple is a power of 2 or a power of 2 times a product of distinct Fermat primes. So an example that establishes that the answer to the question is no can be obtained by choosing $a$ and $b$ such that $atan2(a,b)$ is a rational multiple of $\pi$, say, $z = 1 + i$ so that $atan2(a,b)=\pi/4$, and choosing a value for $n$ such that the resulting rational multiple does not satisfy the criterium described above, like $n=7$.

It would still be interesting to have a test for whether it is possible to denest $i$ in a given radical expression. An algorithm exists for finding a radical expression of minimal nesting depth with the same value as an arbitrary nested radical. Depending on how close to unique the result can be shown to be, this algorithm might suffice to answer the question.