Field Extension Question: $K(a,b) \ne K(a+b)$

Question

I am trying to come up with an example of when an extension $K(a,b)$ with $a,b$ in $E$ is not equal to $K(a+b)$.

In short,

$$K(a,b) \ne K(a+b).$$

I am learning abstract algebra for the first time, and don't even know any explicit Galois theory yet, although the next section in the book will talk about it. Book: Essentials of Modern Algebra Author: Cheryl Chutes Miller. We are in chapter 10, section 1 (if anyone has the same book), and know pretty much everything before that.

Answer 1

$\mathbb{R}(i,-i) = \mathbb{C}\ne\mathbb{R}(i+(-i)) = \mathbb{R}$.

Answer 2

A less trivial example: $\mathbb Z_2(X^2,Y^2)\ne\mathbb Z_2(X^2+Y^2)$.