Galois Theory Through Exercises pp 43-45 | Cite as

# Normal Extensions

## Abstract

In this chapter, we look at the splitting fields of polynomials and emphasize one important property of such fields. This property is contained in the definition of a normal extension: An extension of a field is normal if every irreducible polynomial over this field with one zero in the extension has already all its zeros in it. This means that the extension contains a splitting field of any irreducible polynomial which has at least one zero in it. This definition works equally well for any field extension (also infinite), but we focus here on finite extensions and find that normal extensions and splitting fields of polynomials form exactly the same class. We further discuss a normal closure of a finite field extension. Galois extensions are those which are normal and separable. The separable extensions are discussed in the next chapter.

## References

- [D]R.A. Dean, A rational polynomial whose group is the quaternions, Am. Math. Monthly, 88(1981), 42–45.MathSciNetCrossRefMATHGoogle Scholar