My apologies if this question is somewhat vague or too broad.
Analytically, the fact that $\sqrt{2}$ is no trivial fact. It requires some kind of completion of the rational numbers, e.g. by adjoining the limit of some Cauchy sequence of rational numbers or taking appropriate Dedekind cuts.
Algebraically, one need not accept the existence of limits of Cauchy sequences to talk about $\sqrt{2}$. Instead we talk about adjoining to the rationals the solutions to the polynomial $x^2-2$. Then we can discuss "irrational" numbers in terms of field extensions.
My question is, in what sense are these two points of view equivalent? In both cases we are completing the rational numbers by adjoining extra elements to them, but can one go from one interpretation to another without loss of information? Does every irrational limit of a Cauchy sequence of rationals correspond to a field extension of the rationals, and vice versa?