# Concept of field of roots vs decomposition field (splitting field): what is the difference?

I'm studying in portuguese and here we have the concepts of "field of roots" and "decomposition field". I've been able to find in wikipedia that "decomposition field" stands for "splitting field". I thought that "splitting field" meant "field of roots" but now I'm confused.

Here are two exercises, one using the concept of "field of roots" and the other "decomposition field". Can you guess how different "field of roots" is from "decomposition field'?

Exercise 1)

Find the degrees of the root fields (or field of roots) of the following polynomials over $\mathbb{Q}$:

a) $x^4+1$

b) $x^6+1$

c) $x^4-2$

d) $x^5-1$

Exercise 2) Find the decomposition field (splitting field) of the polynomial $x^4+4$ over $\mathbb{Q}$

Based on these two questions, can you guess what "field of roots" means? It comes before "decomposition field" in my list of exercises. If splitting field is the field where the polynomial splits into linear factors, that is, the field with all its roots adjoined, then what is a "field of roots"?

• At first glance, these look like the same thing. (In Algebraic Number Theory, though, “decomposition field” means something quite other.) – Lubin Sep 24 '17 at 23:25