# Is $\Bbb R$ a splitting field over $\Bbb R$? Over $\Bbb Q$? What does this mean?

There are two problems in Fraleigh's text on abstract algebra.

Which are true?

1.$\mathbb{R}$ is a splitting field over $\mathbb{R}$
2.$\mathbb{R}$ is a splitting field over $\mathbb{Q}$

I could not understand the questions. I know the definition of a splitting field of a polynomial over some field but here, no polynomial is mentioned.

Can someone help to understand the problem and the process of solving these types of problem?

• What does being a splitting field over another field mean? It is either a splitting field for a set of polynomials over some field or else a normal extension.... – DonAntonio Jun 27 '13 at 10:30

1. Can you think of polynomial with coefficient in $\mathbb{R}$ such that all of its zeros are in $\mathbb{R}$ as well?
2. The splitting field $L$ of a set of polynomials in $K[x]$ is gotten by adjoining all the zeros of those polynomials. Consequently $L/K$ is an algebraic extension.
• So I interpret the phrase "a splitting field over $K$" to mean "the splitting field over $K$ of some set of polynomials in $K[x]$." The task for you is to describe such a set, or prove that no such set exists. – Jyrki Lahtonen Jun 27 '13 at 10:44
• @JyrkiLahtonen and ghotan: if you followed my comments below the other answer you know then that Fraleigh considers fields to be candidates for splitting fields over another field $\,k\,$ only those that are contained in $\,\overline k\,$...this automatically rules out $\,\Bbb R/\Bbb Q\,$....weird indeed! – DonAntonio Jun 27 '13 at 10:54