Prove the following a If x0 is a root of a polynomial with c

Prove the following: (a) If x0 is a root of a polynomial with coefficients in then is a root of a polynomial with coefficients in T. (b) Every constructible number is algebraic. (c) The set of constructible numbers is countable. (d) There is a circle with center at the origin that is not constructible.

Solution

(a)

Given (a_n + b_n r) x₀ⁿ + (a_n-1 + b_n-1 r) x_o^n-1 + ... +(a₁ + b₁ r)x₀ + a₀ + b₀ r =0

with a_i , b_i F.

By Dividing through by an + b_n r, we can assume it is poly monic. (ie. coefficients of x₀ⁿ is 1).

x₀ⁿ (a_n-1 + b_n-1 r) x_o^n-1 + ... +(a₁ + b₁ r)x₀ + a₀ + b₀ r =0

Then, x₀ⁿ + a_n-1 x_o^n-1 +....+ a₁x₀ + a₀ = r ( b_n-1x_o^n-1+ ... + b₁x₀ + b₀)

Square both Sides (x₀ⁿ + a_n-1 x_o^n-1 +....+ a₁x₀ + a₀)² - r (b_n-1x_o^n-1 + ... + b₁x₀ +  b₀)² =0

Note: x₀ algebraic if p(x₀)= 0 some polynomial p with rational coefficients.

(b)

Suppose x0 is constructible.

There exists Q = F₀ F₁ F₂ ...... F_k with F_i = F_i-1 (r_i ) some r_i F_i-1 , r_i = 0, r_i F_i-1 such that x₀ F_k

Fk = F_k-1 ( r_k ) x₀ = a + b r_k, a, b F_k

x₀ root of x – ( a + b r_k  ) = 0.

x₀ root of polynomial with coefficients in F_k-1

x₀ root of polynomial with coefficients in F_k-2

x₀ root of polynomial with coefficients in Q.