help with field notation.

Question

I'm doing my undergraduate senior capstone on Schanuel's conjecture. Currently, I am researching James Ax's paper on the conjecture. In his paper he proves Schanuel's conjecture over the formal power series which says let $y_1, \dots, y_n \in t\mathbb{C}[[t]]$ be $\mathbb{Q}$-linearly independent. Then $$\dim_{\mathbb{C}(t)}\mathbb{C}(t)(y_1,\dots,y_n,e^{y_1},\dots, e^{y_n}) \geq n $$ here the $\dim_{\mathbb{C}(t)}$ is the transcendence degree. My question here is two fold. First, What does $t\mathbb{C}[[t]]$ mean? I know $ \mathbb{C}[[t]]$ is the formal power series over the complex numbers, but that extra $t$ in front is throwing me. My other question is how to say all this notation out loud? Especially that long field notation in the conclusion of the theorem. I have a presentation on this and I am completely self taught through books so I don't know how to speak most of this stuff and I don't want to make a fool of myself at the presentation. If somebody could help I would appreciate it. Thank you.

Answer 1

When $R$ is a ring, and $a$ is an element of $R$ then

$$ aR = \{ax : x \in R\} $$

Examples:

• $2 \mathbb{Z} = \{2x : x \in \mathbb{Z}\} = $ the set of even integers
• $t \mathbb{C}[[t]] = \{tf(t) : f(t) \in \mathbb{C}[[t]]\} = \{ \sum_{k \ge 1} a_k t^k : a_k \in \mathbb{C} \} = $ the set of power series which are divisible by t or which have no constant term

Alternative notations for $aR$ include:

• $(a)$
• $\langle a \rangle$

It is called the "principle ideal (of $R$) generated by $a$."

$\dim_{\mathbb{C}(t)}\mathbb{C}(t)(y_1,\dots,y_n,e^{y_1},\dots, e^{y_n}) \geq n$

I would say "the transcendence degree of this extension is at least $n$." You could give a name to "this extension" too, like define $K = \mathbb{C}(t)(y_1,\dots,y_n,e^{y_1},\dots, e^{y_n})$ or something.

$y_1,\dots,y_n \in t\mathbb{C}[[t]]$

Probably I would say "y one through y n are in t-c-t." You can also call $t\mathbb{C}[[t]]$ "the maximal ideal of $\mathbb{C}[[t]]$." Note: $\mathbb{C}[[t]]$ has only one maximal ideal.