Question

If $\mathbf{f}\mathbf{}\left(x\right)\mathbf{\in }\mathbf{R}\left(x\right)$prove that R or C is a splititing field $\mathrm{f}\left(\mathrm{x}\right)$ over R

Short answer

R or C is splitting field.

Step-by-step solution

Definition of splitting field.

A splitting field of a polynomial in a field is a smallest field extension of that field over which the polynomial splitting or split into linear factor.

Showing that R or C is splitting field.

Let $\mathrm{f}\left(\mathrm{x}\right)\in \mathrm{R}\left(\mathrm{x}\right)$. Since $\mathrm{f}\left(\mathrm{x}\right)\in \mathrm{R}\left(\mathrm{x}\right)$therefore the roots of the polynomial $\mathrm{f}\left(\mathrm{x}\right)$will be either a complex number or a number that will belong to R.

If all the root of the polynomial belong to the complex number then splititing field of $\mathrm{f}\left(\mathrm{x}\right)$will be in C and if all the root of the polynomial $\mathrm{f}\left(\mathrm{x}\right)$belong to the real number then splititing field of the polynomial $\mathrm{f}\left(\mathrm{x}\right)$will be in C .

Thus $\mathrm{f}\left(\mathrm{x}\right)\in \mathrm{R}\left(\mathrm{x}\right)$can have two splititing field that is either R or C depends where the roots of the polynomial lies.

Therefore R or C is splititing field over $\mathrm{f}\left(\mathrm{x}\right)$over R.