1. The problem statement, all variables and given/known data Prove that the set of all algebraic numbers is a countable set. Solution: Algebraic numbers are solutions to polynomial equations of the form a_0 x^n + a_1 x^(n – 1) + . . . + a_n = 0 where a_0, a_1, . . . , a_n are integers. Let P = |a_0| + |a_1| + . . . + |a_n| + n . For any given value of P there are only a finite number of possible polynomial equations and thus only a finite number of possible algebraic numbers. Write all algebraic numbers corresponding to P = 1, 2, 3, 4, . . . , avoiding repetitions. Thus, all algebraic numbers can be placed into 1-1 correspondence with the natural numbers and so are countable. The problem along with the solution are also given in the TheProblemAndSolution.jpg file attached. 2. Relevant equations P = |a_0| + |a_1| + . . . + |a_n| + n a_0 x^n + a_1 x^(n – 1) + . . . + a_n = 0 3. The attempt at a solution I'm stuck at the part of the solution that says P = |a_0| + |a_1| + . . . + |a_n| + n. My problem is not just with the computation; I have no idea what P even represents. I'm assuming the letter P was chosen because it's the first letter of the word "polynomial", but then why is that equation different than that of the equation above it that equals zero? If someone could please explain to me what is going on there, it would be GREATLY appreciated!