Field Definition (expanded) - Abstract Algebra

The field is one of the key objects you will learn about in abstract algebra. Fields generalize the real numbers and complex numbers. They are sets with two operations that come with all the features you could wish for: commutativity, inverses, identities, associativity, and more. They give you a lot of freedom to do mathematics similar to regular algebra. Today we motivate the definition of a field by looking at 6 different groups, give the formal definition, and talk about the characteristic of the field and the starting point for all fields - prime fields.

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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition

Milne, Algebra Course Notes (available free online)

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Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro
Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison
Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison

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[NoActuallyGo-KCUF-Yourself]

This might have been the clearest explanation of rings and fields I've seen. Great vid!

[streampunksheep]

Best companion to self learning mathematicians.

[neilclay]

The quality of your teaching is way beyond the average

[ishikasharma]

My problem with abstract algebra has always been intuition, which most professors and videos on the internet skip. I've been through many videos of Socratica's abstract algebra playlist and my basics are so much better! You've given me a simple intuitive approach that I can easily build upon with my textbooks. Special mention to this video, it's eye opening. Thanks for clearing the fog and making abstract ideas so comprehensible. This is rare, keep going, lots of love and gratitude 🙌🏻❤️✨

[nic]

Two days of reading books trying to understand this topic, and this video helps to break down and clear up any misunderstandings in less than 10 minutes. Thank you so much and please never stop making these explanation videos. :)

[vidanatural_oficial]

I love Socratica too.. it is everything that a good channel should be.

[michaelnovak]

Yeay math. Please do videos on topology, real analysis and just any pure math subject you like.

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I like that your teaching videos are short and snappy. I’m extending my maths beyond the applied stuff I learned when studying electronic engineering decades ago. Purely out of whimsical interest and I get a bit addicted to it.

[dinny]

you have explained one of the most difficult math topics and made it look easy. I wish you were my prof in University

[andreibacila]

This is amazing. It took me 30 seconds of watching this video to understand what i have been taking for granted in high school

[webdeveloper-vyhb]

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[spiderjuice]

I'm building a computer and get to choose what instructions it will perform. While watching this video, I realised that I could free up 'space' for one extra instruction (a useful one that previously could not be included) by deleting all of the subtraction-based instructions and instead implementing negation-based instructions to go along with the pre-existing addition-based ones. In effect, I can do everything I could do before, and also got a bonus instruction into the bargain! I just have to perform subtractions in 2 steps instead of 1:
1) negate B
2) add A, B
Credit where it's due: I had the thought to do this when you spoke about additive inverses, so thank you :)

[krishnasharma-hitr]

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[MatheusSouza-bfud]

A good and fun video that we can watch smiling from beginning to end

[Goejii]

"Additive inverse" = "opposé" in french
and "multiplicative inverse" is simply "inverse"

[rogerwilcoshirley]

Excellent topic overview for those of us trying to get started with this and already the door is opening to a much more expansive beautiful intellectual view.

[anusha]

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