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. Here is a link to Gauss's fragment, https://gdz.sub.uni-goettingen.de/id/PPN236010751?tify={%22pages%22:[36],%22panX%22:0.368,%22panY%22:0.464,%22view%22:%22export%22,%22zoom%22:0.498} . 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). I already posted this question on HSM stack exchange, and i'm aware my question is a little bit historical, but i think that the mathematical nature of this question makes it appropriate to post it here.
Therefore, my question is about giving a rough outline and interpretation of the meaning of Gauss's statements in this note.
 A: Despite not understanding a lot of German, I think I get the general idea. Do you know how a real matrix of the form:
$$A = \left( \begin{array}{cc} a & -b \\
                               b & a \end{array} \right) $$
can be thought of as the complex number $a+ib$? It is a famous trick. Essentially, multiplying by $a+ib$ amounts to left multiplying by the real matrix $A$, when you think of each complex number as a column vector consisting of its real and imaginary part.
Well, I think Gauss is using a similar trick in that note, but using the field obtained from $\mathbb{Q}$ (or is he using $\mathbb{R}$?) by adjoining a $5$-th root of unity $\epsilon$.
Then each element in $\mathbb{Q}[\epsilon]$ can be written as $a+b\epsilon + c \epsilon^2 + d \epsilon^3 + e \epsilon^4$, or can be thought of as a $5$-dimensional column vector containing the components $a,\ldots,e$. Multiplying by such an element in that field can be then written as multiplication by a $5$ by $5$ rational (or real if he is using $\mathbb{R}$ as the ground field) matrix satisfying the cyclic conditions you have mentioned. I think this is the basic idea.
To solve the equation $ax = b$ in $\mathbb{Q}[\epsilon]$, where $b \neq 0$, all one has to do is multiply by $a^{-1}$. The unique solution is then $x = a^{-1} b$. The auxiliary equation $ay = 1$ has the unique solution $y = a^{-1}$. And so by solving the auxiliary equation, you are actually getting $a^{-1}$. Then the unique solution of the original equation is then just $x = a^{-1} b$, but you have to represent, either $a^{-1}$ or $b$ (since multiplication here is commutative) as a $5$ by $5$ matrix and the other one by a $5$-dimensional column vector. I think this is what Gauss is doing in that note.
Towards the end of that note, I think that Gauss is calculating $a^{-1}$ (using my notation) using the norm, if I am not mistaken. Let us consider a simpler case. If $z = x + i y \in \mathbb{C}$, then its norm $N(z) = z \bar{z} = x^2 + y^2$. So if $z \neq 0$, then $z^{-1} = \bar{z}/(x^2+y^2)$.
I think what Gauss is doing is something similar. One can write $N(a)$, if $a \in \mathbb{Q}[\epsilon]$, as the product of $a$ and its "conjugates" (over the ground field $\mathbb{Q}$), in the Galois theory sense (by definition of the norm). Please look up the resulting formula somewhere (I do not know the formula of the norm of an element in this algebraic number field). My guess is that, if $a \neq 0$, then $a^{-1}$ should be the product of the Galois conjugates of $a$, other than $a$ itself, divided by $N(a)$, which should be given by a formula, somewhere in the literature. I think this is what Gauss is doing. I hope this is enough details.
Edit: Regarding the last formulas in the article, it seems that Gauss did a "translation" too by what he denotes by $n$. So it is a little bit more complex than what I wrote.
