Let \(F\subset K\) be a field extension. The authors prove a variety of results on Witt rings and Galois cohomology of the ``going-down'' type, i.e. how the behaviour of \(K\) influences that of \(F\). As usual, \(H^ n(F,-)\) denotes the cohomology of the Galois group of a separable algebraic closure of \(F\) and \(F_ q\) the quadratic closure. The projection condition is said to hold for \(F\) in degree \(m\) if the natural morphism \(H^ m(F,\mathbb{Z}/2^ s\mathbb{Z}(m))\to H^ m(F,\mathbb{Z}/2\mathbb{Z})\) is surjective for every \(s\geq1\), where \(\mathbb{Z}/2^ s\mathbb{Z}(m)\) is the \(m\)-th Tate twist of the Galois module \(\mathbb{Z}/2^ s\mathbb{Z}\). Theorem: Let \(F\) be nonreal and \([K:F]<\infty\). Suppose that the projection condition holds in degree \(n-1\) for every finite 2-extension of \(F\) and that \(H^ n(F_ q,\mathbb{Z}/2\mathbb{Z})=0\). Then \(H^ n(K,\mathbb{Z}/2\mathbb{Z})=0\) implies \(H^ n(F,\mathbb{Z}/2\mathbb{Z})=0\). As a Corollary the authors show that \(H^ n(K,\mathbb{Z}/2\mathbb{Z})=0\) implies \(H^ n(F,\mathbb{Z}/2\mathbb{Z})=0\) for \(n\leq3\). Furthermore, under the same hypotheses on \(F\) and \(K\), suppose that the projection condition holds in degree \(n-1\) for every finite 2-extension of \(F\) and for \(K_ q\). Then \(H^ n(K,\mathbb{Z}/2\mathbb{Z})\to H^ n(K_ q,\mathbb{Z}/2\mathbb{Z})\) injective, implies the same injectivity for \(F\). Inter alia, the authors deduce that \(I^ nK=0\) implies \(I^ nF=0\) for \(n\leq4\), where \(IF\) is the fundamental ideal of the Witt ring of quadratic forms over \(F\). They also show that \(K_ n(K)\) 2-divisible implies \(K_ n(F)\) 2-divisible, for \(n\leq4\), and if \(K\) is a finitely generated extension field of transcendence degree \(r\), \(I^{n+r}K=0\) implies \(I^ nF=0\), for \(n\leq4\). Finally, they prove another theorem from which it follows that if \(F\) is any field and \([K:F]<\infty\), then \(I^ nK\) torsion free implies \(I^ nF\) torsion free, for \(n\leq4\), and that if the quadratic forms over \(K\) are classified by the classical invariants, then the same is true over \(F\).