Algebraic Numbers

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

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 \( (2 + 3^{\frac 1 2 } + 5^{\frac 1 2 } )^{\frac 1 2} \).