Local rings k[[T]] and k{{T}}

Question

I have a question of algebraic geometry: A local ring is a ring having a unique maximal. Prove that rings k[[T]] and k{{T}} are local. What are their maximal ideals ?

I see how to manage the fact that the ring of formal series k[[T]] is local, but not the ring of the Puiseux series k{{T}}. I need help to see this last part.

Thanks

Answer 1

At least in the presentations I've seen, $k\{\{T\}\}= \bigcup_n k((T^{1/n}))$ is a union of fields, so is itself a field by definition, and then automatically a local ring.

If you're being asked to prove this in the same context as the fact that $k[[T]]$ is local, presumably it is not safe to take this definition as automatic (i.e., maybe you have $k\{\{T\}\}$ defined using some explicit conditions on the powers of $T$ permitted to appear). But now that you know the unique maximal ideal of $k[[T]]$ to be generated by $T$, you should think about the structure of its fraction field $k((T))$ as far as the powers of $T$ that will be appearing (and similarly for the isomorphic situation of $k[[T^{1/n}]]$ with its unique maximal ideal $(T^{1/n})$ and fraction field $k((T^{1/n}))$) and show that the union I've written above really agrees with the definition you've been given.

Answer 2

Yes, the definition you wrote it's the definition I have. In order to prove that k[[T]] is local I've use this argument: we have this serie $f(T)$=$a_0 + a_1T+ a_2T^2+...$ with $a_0 \not = 0$.

We can find $g(T)$ such that $f(T)·g(T)=1$ So we have some equations, like $a_0b_0=1$, $a_0b_1+a_1b_0=0$... so I can calculate each $b_i$. So any serie with non-zero constant term is a unit, and $(T)$ is its maximal ideal.

I'm not sure if I can repeat the same argument but now with the serie $f(T)=q(T^{1/r})= a_0+ a_1T^{1/r}+a_2T^{2/r}+...$