Timeline for algebraic numbers of degree 3 and 6, whose sum has degree 12

3 events

when toggle format	what		by	license	comment
May 14, 2012 at 18:27	comment	added	KConrad		@Ewan: Your comment isn't another way, by itself, to see the degree is 12, since the degree of $a$ over ${\mathbf Q}(b)$ does not on its own tell you the degree of $a+b$ over ${\mathbf Q}$, which is what your question was about. The point is that ${\mathbf Q}(a+b)$ need not be ${\mathbf Q}(a,b)$. For example, let $r$ and $s$ be two roots of $x^4+8x+12$. Both $r$ and $s$ have degree 4 over ${\mathbf Q}$ and $r$ has degree 3 over ${\mathbf Q}(s)$, but $r+s$ has degree 6, not 12, over ${\mathbf Q}$. In your case, ${\mathbf Q}(a+b) = {\mathbf Q}(a,b)$, but that needs to be checked separately.
Jul 2, 2010 at 8:40	comment	added	Ewan Delanoy		Another way to see it is that $X^6-2$, the annihilator of $a$, factors as $(X-b^2)(X^2+b^2X-b^4)$. The minimal polynomial of $a$ over ${\mathbb Q}(b)$ is $X^2+b^2X-b^4$. So the degree of $a$ over ${\mathbb Q}(b)$ is 2.
Jul 1, 2010 at 6:33	history	answered	KConrad	CC BY-SA 2.5