Question

Show that every element of C is algebraic over R .

Short answer

Cis algebraic over R.

Step-by-step solution

Definition of field extension.

Let K an extension field of F , then K is said to be algebraic extension of F if every element of K is algebraic over F.

Showing that C is algebraic over  .

Let $a+ib\in C$ be an arbitrary element and a polynomial $f\left(x\right)=(x-(a+ib\left)\right)(x-(a-ib\left)\right)$.

Factor theorem: $\alpha$ is a root of polynomial p(x)if and only if $x-\alpha$ is a factor of polynomial p(x).

Thus by this theorem a+ib is zero of f(x) .so,

$$\begin{gathered} f\left(x\right)=(x-(a+ib\left)\right)(x-(a-ib\left)\right) \\ =x^2-(a-ib)x-(a+ib)x+a^2+b^2 \\ =x^2-(a+ib+a-ib)x+a^2+b^2 \\ =x^2-\left(2a\right)x+a^2+b^2 \end{gathered}$$

All the coefficient of polynomial are real. Therefore $f\left(x\right)\in R\left[x\right]$

Thus $a+ib$is a root of a polynomial $f\left(x\right)\in R\left[x\right]$ over R. This implies $a+ib$ is algebraic over R . As $a+ib$is arbitrary element of C.

Therefore every element of C is algebraic over R.