Is there a formula in radicals for the eigenvalues, or at least the largest eigenvalue, of an $n\times n$ symmetric matrix, in terms of the entries? For what $n$ is there a formula? There obviously is if $n\leq 4$, and one suspects not if $n\geq 5$. However, the characteristic polynomial of a symmetric matrix is rather special (e.g. all roots are real) and could possibly be solvable by radicals for larger $n$, although I would guess not.



Is there an algebraic formula for the eigenvalues of a symmetric $n\times n$ matrix?
If I calculated correctly, $$\begin{pmatrix}2 & 1 & 0 & 0 & 0\newline 1 & 3 & 1 & 0 & 0\newline 0 & 1 & 1 & 1 & 0 \newline 0 & 0 & 1 & 1 & 1 \newline 0 & 0 & 0 & 1 & 1 \end{pmatrix}$$ has characteristic polynomial $-x^5+8x^4-20x^3+15x^2+4x-5$ which is irreducible mod $3$ and has $3$ linear factors and a quadratic factor mod $79$, so its Galois group is $S\_5$, so its roots can't be expressed with just radicals.


