# Fermat's theorem for other numbers than whole numbers

Does anyone know what is the status of solutions of Fermat's equation x^n+y^n=z^n for x,y,z other numbers such as 1) integers, 2) algebraic numbers, specially Q[i] and Q(i), complex numbers?

In which numbers does the equation have solutions?

• I can promise you that this is open. Fermat's Last Theorem was barely proven. Commented Nov 16, 2019 at 7:35
• Well, some of these are not open - the algebraic numbers are an algebraically closed field, so any polynomial has the correct number of roots. Integers reduces to the case of natural numbers: if $n$ is even, just flip the signs on the negative integers, if $n$ is odd, just move the negatives integers to the other side. The other case of $\Bbb Q[i]=\Bbb Q(i)$ I do not immediately know the answer to, though. Commented Nov 16, 2019 at 7:59

$$2^3+(-2)^3=0^3$$ (for integers)
$$1^4+0^4=i^4$$ (complex)
$$(2^{\frac{1}{3}})^3+(5^{\frac{1}{3}})^3=(7^{\frac{1}{3}})^3$$ (algebraic)
• The Gaussian integers may be more interesting if $0$ is not allowed. Commented Nov 16, 2019 at 9:59