# Interpretation of a short note of Gauss on the resolution of a special system of inhomogeneous linear equations by roots of unity

My question refers to a 2-pages fragment of Gauss, entitled: "Note on the resolution of a special system of linear equations", which is found on pages 30-31 of volume 8 of his works. In this fragment, Gauss expresses the solution vector $$(P,Q,R,S,T)$$ of the following inhomogenous system of $$5$$ linear equations:

$$aP + bQ + cR + dS + eT = A$$ $$bP + cQ + dR + eS + aT = B$$ $$cP + dQ + eR + aS + bT = C$$ $$dP + eQ + aR + bS + cT = D$$ $$eP + aQ + bR + cS + dT = E$$

(note that the coefficents of each row are a cyclic shift of the previous row) by five auxillary parameters $$p,q,r,s,t$$ which solve another system of $$5$$ linear equations:

$$ap + bq + cr + ds + et = 1$$ $$bp + cq + dr + es + at = 0$$ $$cp + dq + er + as + bt = 0$$ $$dp + eq + ar + bs + ct = 0$$ $$ep + aq + br + cs + dt = 0$$

Gauss then expresses $$P,Q,R,S,T$$ through a bilinear expression in $$p,q,r,s,t, A,B,C,D,E$$. To find $$p,q,r,s,t$$ Gauss employs a certain algebraic trick involving the fifth root of unity $$\epsilon$$ ($$\epsilon^5 = 1$$).According to the short comment of Fricke on Gauss's result, the final form of Gauss's formulas for $$p,q,r,s,t$$ involves an expression for the norm of the cyclotomic number $$a +b\epsilon + c\epsilon^2+d\epsilon^3+e\epsilon^4$$ in the quintic (cyclotomic) field $$Q[\epsilon]$$ (see also this post: https://math.stackexchange.com/questions/24840/finding-the-norm-in-the-cyclotomic-field-mathbbqe2-pi-i-5 ). The whole Gaussian method looks like an hybrid between techniques of linear algebra and techniques fron alegraic number theory (roots of unity), and therefore it aroused my curiosity (this complicated procedure suggests that Gauss had an idea here...). Here i omitted the final formulas, because they are long and the best way to gain insight into them is simply to look into Gauss's note.

I understand that trying to give a sound interpretation of such a complicated note (and without supporting context) is a very difficult task, but maybe someone who is familiar with advanced algebraic techniques will find familiar patterns in Gauss's fragment (honestly i have no clue or idea about how to dechiper this note).

Therefore, my question is about giving a rough outline and interpretation of the meaning of Gauss's statements in this note.