The title sais it already:
Why is $L=\mathbb{Q}(\sqrt[1]{2})\cup\mathbb{Q}(\sqrt[2]{2})\cup\mathbb{Q}(\sqrt[3]{2})\cup\cdots$ a field?
The hint provided in my textbook is: $\mathbb{Q}(\sqrt[n]{2})\cup\mathbb{Q}(\sqrt[m]{2})\subset\mathbb{Q}(\sqrt[mn]{2})$, but this doesn't really get me anywhere. Actually, I have no idea what to do whatsoever. Could anyone clarify or give some hint please?