The Primitive Element Situation is Crazy

[@AsiccAP]

love this video! I do hope the video is a bit louder though

[@sw3aterCS_]

6:38 It’s clear to me that the elements shown here span the field, but how do we know that they are Q-linearly independent?

[@Liam-tv4ne]

What sort of software did you use for the visualizations for the video? 3B1B python library?

[@DrMcCrady]

Yep.

[@stephenliao63]

This video has a volume that I can’t barely hear with hardware volume maxed out, and suddenly an ad showed up, That was ear blowing.😂

[@DrMcCrady]

Sorry about that!

[@asmithgames5926]

So what does the theorem say? If you add any finite number of extensions, that you can always represent then with a single element extension that is often just their sum?

[@asmithgames5926]

Ok I got it... In field theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem implies in particular that all algebraic number fields over the rational numbers, and all extensions in which both fields are finite, are simple.

[@DrMcCrady]

The theorem statement is that all finite, separable extensions are simple. The statement of the theorem does not give a recipe for how to construct a primitive element. The proof of the theorem gives such a recipe. planetmath.org/proofofprimitiveelementtheorem

[@moonmartex]

Pretty dope theorem

[@asmithgames5926]

Good video. A little rushed. Volume is too low!

[@kent_hdd]

You need to do a little work on your audio editing skills. Other than that, great video!

[@DrMcCrady]

Thanks, I’ll figure it out!

[@firstlast-wg2on]

@@DrMcCrady grab a compressor, dial the threshold down to the volume of when your voice is quietest, and then dial the ratio up to decrease the distance between the highest volume and lowest volume parts. Adjust attack to be faster if you feel your voice is popping to loudly initially, and increase your release on the compressor based on how “smooth” you want the audio to sound (but bear in mind, it can sound unnatural and like an overall flat volume decrease if the release is too low). Throw a de-easer on top and pick out the 3k+ range to cut. Those two things generally are enough.