Algebraic Numbers

Algebra Level 5

A number is called algebraic if there is a polynomial with rational coefficients for which the number is a root. For example, 2 \sqrt{2} is algebraic because it is a root of the polynomial x22 x^2 - 2 . The number 2+3+5 \sqrt{ 2 + \sqrt{3} + \sqrt{5} } is also algebraic because it is a root of a monic polynomial of degree 8, namely x8+ax7+bx6+cx5+dx4+ex3+fx2+gx+h. x^8 + ax^7 + bx^6 + cx^5 + dx^4 + ex^3 + fx^2 + gx + h. Find a+b+c+d+e+f+g+h. |a| + |b| + |c| + |d| + |e| + |f| + |g| + |h|.


Details and assumptions:

  • A monic polynomial is a polynomial whose leading coefficient is 1.
  • For those who can't see the square roots clearly, the number is (2+312+512)12 (2 + 3^{\frac 1 2 } + 5^{\frac 1 2 } )^{\frac 1 2} .
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