Hi thanks for your question. My name isXXXXX can answer this.
There is a right to written reasons for the dismissal, under s.92 of the Employment Rights Act 1996.
However, it also states that this right is only exercisable once you have worked there for one year or more.
Sadly, therefore, there is no right to demand this statement if you've been there for one month.
The only way you're likely to get it, is by making a complaint to the tribunal saying what you believe the reasons for dismissal were, and they would have to confirm or deny it. It's not a proper use of the tribunal procedure, especially if they've paid you the notice period due under the contact, but it's likely to be the only way you will force them to give this information.
So essentially I should just ask nicely.
How would I go about going the tribunal route if they refuse?
Asking nicely is a good start :)
If they refuse, then you could complaint to your local tribunal: have a look at this - it has links to the online claim system. http://www.justice.gov.uk/tribunals/employment