С днём рождения, kdv2005

[ny_quant]

После некоторых сомнений, решил поздравить именинника нестандартной (как мне кажется) задачей.

Для каких N существуют матрицы размера NxN с рациональными элементами, удовлетворяющие уравнению A^3+A+I=0

Комментарии пока что буду скринить, чтобы всем было интереснее решать.

Comments

[misha-b.livejournal.com]

Sorry, chto-to ya nevnyatno izlagaju.

1. All eigenvalues of the matrix are roots of x^3+x+1=0.
Let's call them a,b,c

2. det A is a product of these eigenvalues and therefore has the form
a^k b^l c^n for some integers k,l,n. Clearly det A is a rational number (all coefficients of the matrix are rational).

3. Let us now consider a polynomial of three variables P(x,y,z).
Claim:
P(a,b,c) is rational if and only if P can be written as a sum
(with rational coefficients) of powers of the elementary symmetric polynomials xyz, xy+xz+zy, x+y+z.

4. Consider now P(x,y,z)= x^k y^l c^n . We know that P(a,b,c) is rational.
Therefore P is a sum of powers of the elementary symmetric polynomials.
By some additional simple argument, which I omit, the only polynomial
which can appear is xyz. Therefore, it is a power of xyz and roots go in triples.

Mne kazhetsya, primerno tak.