# Zulip Chat Archive

## Stream: new members

### Topic: Formalising towers of field extensions

#### Jakob Scholbach (Feb 26 2021 at 19:59):

Another newbie question: What are efficient ways of encoding this common situation in field theory: "Let K \subset E \subset F be a tower of field extensions. If x in K is separable over F, then x is separable over E." When I try to formalise this, I find it cumbersome to use the `[algebra K E] [algebra E F]`

notation since then I need to manually compile the composition of these two algebra maps into a new algebra to express the separability of x over K. I would normally think I should start from the datum of a functor (0 \to 1 \to 2) to the category of commutative rings (or fields). Is this finite category in the library? Or are there other convenient ways? Thanks!

#### Adam Topaz (Feb 26 2021 at 20:01):

This is a can of worms which can be tamed with docs#is_scalar_tower

#### Adam Topaz (Feb 26 2021 at 20:06):

So for example you can declare a tower of fields $K \subset L \subset M$ as follows:

```
import algebra.algebra.basic
variables {K L M : Type*} [field K] [field L] [field M]
[algebra K L] [algebra K M] [algebra L M] [is_scalar_tower K L M]
```

#### Adam Topaz (Feb 26 2021 at 20:09):

Concerning your category $0 \to 1 \to 2$, this is "just" `fin 2`

with the category instance obtained automatically from its ordered structure.

#### Jakob Scholbach (Feb 26 2021 at 20:10):

Thanks, @Adam Topaz!

#### Johan Commelin (Feb 26 2021 at 20:11):

@Jakob Scholbach It took a while for us to figure this out. I very much agree that it is cumbersome that we need to write two lines of `variables`

just to say "let $K \subset L \subset M$ be a field extension". But once you've done that at the top of your file, it is quite pleasant to use afterwards.

#### Johan Commelin (Feb 26 2021 at 20:11):

Other variants are short to introduce, but painful to use.

#### Adam Topaz (Feb 26 2021 at 20:12):

I would certainly advise against using a functor from `fin 3`

if you want to do any nontrivial algebra :)

#### Johan Commelin (Feb 26 2021 at 20:12):

There are some ideas how to make the `variables`

line a lot shorter, but they would require modifying lean, and nobody has done that. Now that lean 4 is out, I guess we will wait till mathlib is ported before experimenting with such new features.

#### Jakob Scholbach (Feb 26 2021 at 20:15):

OK, this is at any rate beyond my horizon, so I will use the scalar tower then.

Last updated: May 11 2021 at 23:11 UTC