What does it mean for a field to be generated by some set of elements?

Question

I've read up online, but I'm having trouble understanding what it means for a field $F$ to be generated by a set of elements $S = \{\alpha_1, ..., \alpha_n\}$ over another field $K$.

What are the implications?

Answer 1

Well, let $\Omega$ be some field, thought of a huge, like the complex numbers. Let $K$ be a subfield of $\Omega$, and $S$ a subset of $\Omega$. Then there are two equivalent definitions of $K(S)$.

The first is like unto the explanation given by @EeveeTrainer: $K(S)$ is the set of all rational expressions with $K$-coefficients that involve only finitely many elements of $S$. More explicitly, for $z\in\Omega$ to be in $K(S)$, it means exactly that there must be a finite subset $S_z\subset S$, and a rational expression $$ \frac{f_z(s_1,\cdots,s_{m_z)}}{g_z(s_1,\cdots,s_{m_z})}\,, $$ where $s_1,\cdots,s_{m_z}$ are the elements of $S_z$, and $f_z$ and $g_z$ are polynomials with coefficients in $K$, in $m_z$ indeterminates. I hope I’ve made it clear in the notation that the number of indeterminates depends on $z$, as will the particular polynomials $f$ and $g$.

The cumbersomeness of the above first definition makes the other definition feel like a breath of spring air: $K(S)$ is the intersection of all subfields of $\Omega$ that contain $K$ and $S$.

It’s up to you to prove that the two definitions are equivalent. Having both under your belt makes proving things about $K(S)$ much easier, for sometimes the first definition is more convenient to use, sometimes the second.