splitting field
Eigenvalues are roots of the characteristic polynomial, but roots may not live in the field you started with. The real matrix [0, -1; 1, 0] (a 90-degree rotation) has characteristic polynomial x^2 + 1, which has NO real roots — so over R this operator has no eigenvalues at all. To find them you must enlarge the field. The splitting field is the smallest field extension in which chi(x) factors completely into linear pieces.
Concretely, given chi(x) over a field F, its splitting field is the smallest field containing F and all the roots lambda_1, ..., lambda_n, so that chi(x) = (x - lambda_1)...(x - lambda_n) there. For the rotation above the splitting field is C, where x^2 + 1 = (x - i)(x + i) and the eigenvalues i and -i finally appear.
Why bother? Several theorems only switch on once the characteristic polynomial splits. Triangularization over a field requires chi to split there (Schur's theorem needs C, or any algebraically closed field); the existence of n eigenvalues counted with multiplicity needs splitting; Jordan form is a theorem over an algebraically closed field for exactly this reason. Passing to the splitting field is the standard move that makes these tools available.
An honest caveat: enlarging the field is a genuine change of scenery, not free. A real operator that is irreducible over R may have complex eigenvalues that come in conjugate pairs, and the real canonical forms (real Jordan form, rational canonical form) exist precisely to keep you in your home field when you must. The splitting field is the price of full diagonalizability or triangularizability.
A rotation has no real eigenvalues; only after passing to its splitting field C does the characteristic polynomial split and the eigenvalues appear.
Over an algebraically closed field like C every polynomial already splits, so the splitting field is the field itself — which is why complex linear algebra is so much smoother. The minimal polynomial may stay irreducible over a small field yet break apart over its splitting field.