| 00:00 - 00:03 | ... and now we replace the field with a ring |
| 00:04 - 00:05 | acting on abelian group |
| 00:05 - 00:07 | as a so-called module. |
| 00:08 - 00:12 | Take the integers modulo two as a |
| 00:12 - 00:15 | submodule of the integers modulo six. |
| 00:17 - 00:19 | ... and this "vector space" |
| 00:19 - 00:21 | has some basis and dimension. |
| 00:24 - 00:26 | Not all ... |
| 00:27 - 00:28 | Some modules ... |
| 00:31 - 00:33 | Some modules do not admit even a single basis. |
| 00:34 - 00:36 | Those that do are called free. |
| 00:53 - 00:58 | How can it not admit a basis? Even. the. standard. basis? |
| 01:13 - 01:15 | Over fields it was easy! |
| 01:15 - 01:17 | What happened to the finite-dimensional spaces!? |
| 01:18 - 01:23 | But you say yesterday that having a basis is not obvious! |
| 01:25 - 01:28 | Something about being able to make a choice. |
| 01:29 - 01:31 | And now you say to me ... |
| 01:31 - 01:34 | "existence of a basis fails!" |
| 01:34 - 01:37 | There is something wrong with your working ... |
| 01:37 - 01:40 | I will find it your silly mistake! |
| 01:40 - 01:42 | We make no mistake, please just work it out for ... |
| 01:42 - 01:46 | Why don't we restrict ourselves to free modules? Wouldn't that be better! |
| 01:46 - 01:48 | But why? That it unnecessary. |
| 01:48 - 01:52 | You have, at most, the smarts of a bunch of farts. |
| 01:53 - 01:54 | So I learned ... |
| 01:56 - 01:57 | that only free modules admit a basis. |
| 01:57 - 02:00 | Yeah! But then we have dimension as before. |
| 02:00 - 02:03 | No more of this silly nonsense about no basis! |
| 02:04 - 02:08 | Now can we please get back to the lesson. |
| 02:08 - 02:13 | You have wasted plenty of time so far. Please can you get back to teaching! |
| 02:14 - 02:16 | Learning is my utmost passion. |
| 02:17 - 02:21 | While we were messing around, guess who passed *his* algebra exam? Stalin! |
| 02:27 - 02:29 | What else can you teach me? |
| 02:30 - 02:34 | I'll tell you what. I like the finite fields! |
| 02:34 - 02:36 | Hands-on computation. |
| 02:41 - 02:42 | Alright. |
| 02:43 - 02:47 | When I am a field of prime characteristic, now then I must be finite right? |
| 02:48 - 02:53 | Or do you deny that too? Like you did my basis. |
| 02:54 - 02:56 | I was surprised at first by the prime powers. |
| 02:56 - 02:59 | When were prime fields not enough? Why prime powers! |
| 03:00 - 03:02 | You are insufferable. "Axiom of choice"! |
| 03:04 - 03:07 | Was your dad pro-choice? Did he accept AC? |
| 03:14 - 03:16 | But then again, at least those ... |
| 03:19 - 03:23 | "free modules". At least they have some dimension. |
| 03:25 - 03:26 | That's obvious. |
| 03:31 - 03:33 | Just find a basis. |
| 03:40 - 03:46 | That basis, it has some cardinality. Yeah. |
| 03:46 - 03:49 | That is the dimension. |
| 03:53 - 03:56 | No question. |
