A few historical remarks about algebraic determinations of the sign of the quadratic gaussian sum might not be out of order.

The proof in David's post was first given by Kronecker, according to Hasse's Vorlesungen. The only analytic ingredient is the determination of the sign of the sin function. This proof is reproduced in Fröhlich and Taylor, Algebraic Number Theory, pp. 228--231.

A different algebraic proof, using the same analytic ingredient, was given by Schur and can be found in Borevich and Shafarevich, Number Theory, pp. 349--353.

Hasse's Vorlesungen also contain a proof by Mordell in which the analytic ingredient is replaced by the fact that if a polynomidal $f\in{\mathbf Z}[T]$ has opposite signs at $a,b\in{\mathbf R}$, then it has a root between $a$ and $b$. This can be proved using the purely algebraic theory of Artin and Schreier.

If you are looking for a proof using more analysis, not less, see Rohrlich's survey on Root Numbers in Arithmetic of L-functions, pp. 353--448.

Addendum. A nice (if somewhat dated) survey on The determination of Gauss sums can be found in the BAMS 5 (1981), 107-129. I learnt there that the proof attributed by Hasse to Kronecker actually goes back to Cauchy. New proofs are still being given; see for example Gurevich, Hadani, and Howe, Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation. Int. Math. Res. Not. IMRN 2010, no. 19, 3729–3745, available here.

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A few historical remarks about algebraic determinations of the sign of the quadratic gaussian sum might not be out of order.

The proof in David's post was first given by Kronecker, according to Hasse's Vorlesungen. The only analytic ingredient is the determination of the sign of the sin function. This proof is reproduced in Fröhlich and Taylor, Algebraic Number Theory, pp. 228--231.

A different algebraic proof, using the same analytic ingredient, was given by Schur and can be found in Borevich and Shafarevich, Number Theory, pp. 349--353.

Hasse's Vorlesungen also contain a proof by Mordell in which the analytic ingredient is replaced by the fact that if a polynomidal $f\in{\mathbf Z}[T]$ has opposite signs at $a,b\in{\mathbf R}$, then it has a root between $a$ and $b$. This can be proved using the purely algebraic theory of Artin and Schreier.

If you are looking for a proof using more analysis, not less, see Rohrlich's survey on Root Numbers in Arithmetic of L-functions, pp. 353--448.