# 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.

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.

• See also circulant matrices, for instance on wikipedia, or elsewhere. After rearranging the equations, the $5$ by $5$ matrix becomes a circulant matrix. Jul 5, 2020 at 15:52
• Thanks Malkoun! clearly your answer helps a lot and gives the basic idea of the note (i voted your answer). I didn't accept the answer yet because i still need to process it in my mind and see how the pieces of the puzzle combine. Right now i'm still not sure it's a definitive answer (and obviously it might be a definitive answer, but i need to check i understand all aspects of Gauss's note). Thanks a lot! i'll accept the answer once the subject will be clear to me. Jul 5, 2020 at 15:58
• No worries. If you need more help, let me know. Jul 5, 2020 at 16:02
• To be honest, the note is clear to me, except how he got the solution of the auxiliary system at the end. I mean, one needs to probably compute to check it (sorry, this would take time, so I will not do it). Jul 5, 2020 at 16:27
• this is exactly but i wanted to ask - i followed your answer and clearly saw your interpretation fits exactly to the systems of linear equations that descibe the arithmetic in the quintic field $Q[\epsilon]$, but what i don't understand is the formulas right after the systems (immediately after the systems he states several formulas which connect $p,q,r,s,t$ with $a,b,c,d,e$ ) and at the very end (the formulas that involve the norm of the cyclotomic number). If you won't do it, can you at least give a hint about how to arive the solution at the very end of te note? Jul 5, 2020 at 16:56