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"?

  • 1
    $\begingroup$ At first glance, these look like the same thing. (In Algebraic Number Theory, though, “decomposition field” means something quite other.) $\endgroup$ – Lubin Sep 24 '17 at 23:25

From all the evidence I can find, corpo de raízes ("field of roots") and corpo de decomposição ("decomposition field") are synonyms, and both translate to English splitting field. For instance, I found these notes which define corpo de raízes and Portuguese Wikipedia which defines corpo de decomposição, with both definitions being the usual definition of splitting field. I also can't think of any other meaning that would make sense in context. So it seems these exercises are just mysteriously inconsistent in their terminology.


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