Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 101 / 3 / iii / Solution

Choose $a\in K\setminus F$. The multiplication homomorphismhasThis element is nonzero. Since $1,a$ are linearly independent over $F$, there is an $F$-linear functional $\lambda :K\to F$ with $\lambda (1)=0$ and $\lambda (a)=1$. Applying $\lambda \otimes {\mathrm{id}}_K$ sends the displayed tensor to $1$.

Thus the unital ring homomorphism $\mu$ has a nonzero kernel. A unital homomorphism from a field is injective, so $K{\otimes }_{F}K$ is not a field.