Algebraic proof of non-trivial solution to the Pell's equation Let $d$ be a square-free positive integer, and consider the pairs $(x, y) \in \mathbb{Z}^2$ that satisfy:
$$x^2 - dy^2 = 1$$
The existence of a non-trivial solution to this equation (i.e. distinct from $(\pm 1, 0)$) is equivalent to showing existence of a torsion-free unit in $\mathbb{Z}[\sqrt{d}]$. All proofs I've seen of this fact uses the Dirichlet Approximation Theorem, and then follows by consecutive applications of the Pigeonhole Principle. But the group of units in $\mathbb{Z}[\sqrt{d}]$ is a purely algebraic construction, so I wonder if it's possible to prove this fact without resorting to the approximation theorem. Is there a known "algebraic" proof of this?
 A: Given the Dirichlet Unit Theorem which states that the group of units in the ring of integers of a number field is finitely generated and has rank equal to $r = r_1 + r_2 − 1$, where $r_1$ is the number of real embeddings and $r_2$ half the number of complex embeddings of the field, you're guaranteed the existence of a fundamental unit in the ring of integers of a field $\mathbb{Q}[\sqrt{d}]$ as a real quadratic field has $r_1=2$.  It's simple enough to translate back and forth between units and solutions of Pell's Equation as you've noted.
So to answer your question we would need to look at proof's of Dirichlet's Unit Theorem, and, in fact, most proofs apply Minkowski's Theorem to a logarithmic embedding of the units into a real vector space.  Such proofs wouldn't qualify as "algebraic" proofs, but it is possible to give a more purely algebraic proof of Dirichlet's Unit Theorem by introducing the concept of ideles.  Such an approach can be seen in Weil's Basic Number Theory or in Cassels and Frohlich Algebraic Number Theory where the unit theorem follows from the compactness of a certain idele group called $\mathbb{J}_K^1/K^\times$.
A: Assume that $d = p \equiv 1 \bmod 4$ is a prime number. The field of $p$-th roots of unity contains the real quadratic number field 
$k = {\mathbb Q}(\sqrt{p})$. The norm of the cyclotomic unit $1 + \zeta$ is a unit in $k$, and you will have an algebraic proof of the solvability of the Pell equation if you can show that this unit is not $\pm 1$. This must be east to prove using the sine function (this proof is due to Dirichlet).
If you want to avoid transcendental function, you will have to invoke congruences, but I haven't checked how far this will get you. Things will get more technical in general; for primes $p \equiv 3 \bmod 4$ you already have to look at the field of $4p$-th roots of unity. 
