Does this Xbox one controller work without a cable?

I bought an xbox one controller with cable from Saturn for 50.99 euro.

According to the specification, it should also work with the xbox one without a cable.

Does it also work on a PC without a cable?

The cable doesn't bother me, but I would like to know.

Unfortunately, there's no Windows 10 adapter

But should that work with bluetooth?

I'll try it tomorrow morning, but Microsoft always wants money for adapters. Sign up tomorrow