= Solution
The <eigenvalues> in $K$ are precisely the roots in $K$ of the companion <polynomial> $d$; over an <algebraic closure> take all its roots. Thus the form determines their root set, but \b[an explicit list generally requires factoring or solving $d$ and cannot be read off by inspection]. This is the relevant distinction in the qualification “with no further calculations”. Over a general <field> the <polynomial> may have no roots in $K$ at all, and its diagonal entries are not its <eigenvalues>. If “determinable” means only uniquely specified rather than immediately readable, all the information is of course fixed by the <matrix>, including this root set.
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