Proof of Fermat's last theorem for $n=5$ using primitive roots of unity? I've been reading "An introduction to the theory of numbers" by Hardy and Wright and they gave a nice proof of Fermat's last theorem for $n=3$ by proving that there are no solutions to $$x^3+y^3+z^3=0$$ where $x$,$y$, and $z$ are numbers of the form $a+bp$ where $a$ and $b$ are integers and 
$$p=e^{2/3\pi i}=\frac 12\Big(-1+\sqrt{3}i\Big)$$ which is a primitive third root of unity. The general approach of the proof is to factor $x^3+y^3$ into $$(x+y)(x+py)(x+p^2y)$$
They also then state that the proof of FLT for n=5 is very similar but uses $p=e^{2/5\pi i}$ which is a primitive fifth root of unity. 
My question is, does anyone know where I can find this proof?
I'm looking for a proof of FLT for $n=5$ that uses properties of integers of the form $a+bp$, and specifically the fact that $x^5+y^5$ factors into $$(x+y)(x+py)(x+p^2y)(x+p^3y)(x+p^4y)$$ I don't expect anyone to type up the full proof but a link or reference would be nice.
 A: First of all, in $\mathbb{Z}[\zeta]$ with $\zeta=e^{2\pi i/5}$, elements are given by $a+b\zeta+c\zeta^2+d\zeta^3$. To prove Fermats last theorem, one can distinguish between two cases: 5 doesn't divide $x$, $y$ or $z$, and in the other case $5$ only divides $z$ (if $5$ divides $x$, then switch $x$ and $z$). The second case is more difficult to prove than the first case. Note that $\mathbb{Z}[\zeta]$ is an unique factorisation domain.
The summary is as follows. It can be shown that the given factors in the factorisation of $x^5+y^5$ are pairwise coprime, here is used that $5$ doesn't divide $xyz$. Because $x^5+y^5=z^5$, the prime elements in every factor appear with a fifth power. Because $x+y\zeta=ua^5$ with $u$ a unit and $a\in\mathbb{Z}[\zeta]$, it follows that $x\equiv y\pmod{5}$. Here is used that every unit in $\mathbb{Z}[\zeta]$ is the product of a real unit and some power of $\zeta$. Repeating this with $x^5+(-z)^5=(-y)^5$, it follows that $x\equiv-z\pmod{5}$.
From this follows that $$x^5+y^5\equiv2x^5\equiv(-x)^5\pmod{5}$$ and $3x^5\equiv0\pmod{5}$, from which a contradiction follows.
A good sources for this proof is Number Fields by Daniel A. Marcus. The proof for the second case can be found in Number Theory by Borevich and Shafarevich. If you understand Dutch, you can also read my bachelor thesis about Fermat's last theorem for regular primes, which I'm writing at the moment.
