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

[prof-yura.livejournal.com]

The polynomial x^3+x+1 is irreducible over the rationals; in particular, all its roots are Galois-conjugate over the field Q of rational numbers. Since the degree N characteristic polynomial of the matrix has rational coefficients, all the roots of x^3+x+1 have the samr multiplicities as roots of the characteristic polynomial. This implies that N is divisible by 3.

[misha-b.livejournal.com]

> Since the degree N characteristic polynomial of the matrix has rational coefficients, all the roots of x^3+x+1 have the samr multiplicities as roots of the characteristic polynomial.

How do you see this?

[prof-yura.livejournal.com]

Well, it's similar to the situation with complex-conjugate roots of a polynomial with real coefficients: they have the same multiplicity. Here the roots of x^3+x+1 are Galois-conjugate over the rationals and therefore if one of them has say, multiplicity M as a root of a polynomial f(x) with rational multiplicities then all other roots of x^3+x+1 have the same multiplicity as roots of f(x).

[misha-b.livejournal.com]

Thanks! Of course, it is just needs to be invariant under conjugation.
I was making it far too complicated.

You are welcome

[prof-yura.livejournal.com]

a polynomial f(x) with rational multiplicities

multiplicities => coefficients