Fun fact: since a relatively recent reform the position of the diaeresis (in french: "tréma") kinda changed (at least for quite a few words). For example in the word "aigue", the diacritic sign can be put either on the 'e' (as used to be the case previously) or the 'u' (which is now recommended but not mandatory: you can still put it on the 'e' instead). So both "aigüe" and "aiguë" are correct AFAICT.

Although I'm a native french speaker TIL, thanks to you, that it's role was to inform that the letters are to be pronounced separately.