I am currently beginning a long term investigation (I will have 6 months of time). I wish to better understand fundamental properties of polynomials and eventually come up with a good root finding algorithm.
I really almost know nothing advanced about the subject. But anyhow, I have considered studying towards these four areas:
1 - Learn about field theory and abstract/advance my knowledge about polynomials
2 - learn about existing algorithms - (bracketing and open methods)
3 - some basic abstract algebra (Galois) - group theory
4 - learn about basic complexity theory and be able to analyze algorithms for their efficiency
SO, my question: What are some books and sources you guys recommend, where I can begin my study. Additionally, are my goals feasible or should I redirect my investigation (if you think there are better/more appropriate things to study)?
Do note, I am willing to spend lots of time and understand as much as possible, but my formal education in mathematics is not even at an undergraduate level (but I give it my best shot).