Let \(K\) be an algebraic function field in one variable over an algebraically closed field \(k\) and let \(F\) be a finite extension field of \(K\). A basic theorem of Tsen, the norm theorem, states that every element of \(K\) is the norm of an element of \(F\) [\textit{C.C. Tsen}, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. I, No. 44, II, No. 8, 335--339 (1933; Zbl 0007.29401)]. The authors here give a simpler proof of this result which provides more information, thus they prove the following result, where \(g_K\) denotes the genus of \(K\) and \(N_{F/K}\) the norm. Theorem 1. Given \(F\supseteq K\) as above, where \([F: K]= n\), and put \(m = n(g_F- g_K)\). Then for any \(a\in K\) there exists \(b\in F\) such that \(N_{F/K}(b)= a\), \(b\) has at most \(m\) zeros which do not lie over a zero of \(a\) and at most \(m\) poles not lying over a pole of \(a\). The proof uses properties of holomorphic \(1\)-forms on Riemann surfaces, and at first is only given for the special case \(k= \mathbb{C}\), but is then transferred to the case of algebraically closed fields of characteristic zero by a model-theoretic argument. The authors are also able to the prove the original norm theorem in finite characteristic, but information about the poles of \(b\) is replaced by information about the coefficients of its minimal polynomial.