What is the prerequisite knowledge for a career in quantitative finance?

I've previously attempted to ask this question at the Quantitative Finance SE but it's not allowed, hopefully I can ask this here, if not then i'm sorry.

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I'm in the process of looking at potential careers that I may undertake after one day completing a degree in mathematics (I've yet to start a degree). One that particularly interests me is a career in quantitative finance. It looks like a promising career because you're in a stimulating environment surrounded by other able professionals, it allows you to apply all that you have learnt, and in certain companies you're even encouraged to research and develop your knowledge even further, and it also rewards you quite well.

As I have yet to undertake a degree in mathematics, I'd like to know:

1. The prerequisites mathematical knowledge for a future in such a field: I'm aware that one must have a sound knowledge of PDE's, Probability, Statistics. Could anyone be more specific with what I need to know in this regard, i.e. Markov Chains, etc. I need specifics.


2. Following from this, how should I specialise once I've reached the later years of my degree: Should my degree be statistics/probability skewed towards the end?


3. Is a master's degree in Mathematics the recommended route? Are there diminishing returns with a Phd?


4. I'm aware that I also need to have knowledge of programming, what languages are commonly used in this industry? R, C++? What languages would serve me well?


5. What other skill sets are required?

I apologise in advance if I haven't provided enough details, I do not have enough information to know what I want to do in precise detail in this field yet. Thanks.