A short elementary proof of Grothendieck's theorem on algebraic vectorbundles over the projective line

Let E be an algebraic (or holomorphic) vectorbundle over the Riemann sphere $\text{P}$¹($\text{C}$). Then Grothendieck proved that E splits into a sum of line bundles E = ⊕L_i and the isomorphism classes of the L_i are (up to order) uniquely determined by E. The L_i in turn are classified by an integer (their Chern numbers) so that m-dimensional vectorbundles over $\text{P}$¹$\text{C}$ are classified by an m-tuple of integers

$\text{κ(E) = (κ}{}_1\text{(E),…,κ}{}_{\mathrm{m}}\text{(E)), κ}{}_1\text{(E)≥?≥κ}{}_{\mathrm{m}}\text{(E), κ}{}_{\mathrm{i}}\text{(E)∈}\text{Z}$

.In this short note we present a completely elementary proof of these facts which, as it turns out, works over any field k.