Field Definition (expanded) - Abstract Algebra
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- เผยแพร่เมื่อ 12 ก.ค. 2018
- 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
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Milne, Algebra Course Notes (available free online)
www.jmilne.org/math/CourseNote...
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Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison
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Can you please add videos on Linear algebra.
Moomomomoommoommo
must not a field have a propert that x*y means adding x+x+x...x y - times, or it can be different?
Hello, can you please tell me how to translate equations like (x^2+5x+6=0) into a math field? I tried looking up that in ANY way, but I've been having no luck finding a way. 😓😣😢
This might have been the clearest explanation of rings and fields I've seen. Great vid!
+ Groups
when i took abstract we did not study rings, integral domains or fields in class, were just given 3 pdfs that we were to study before the final exam
Best companion to self learning mathematicians.
Along with Khan Academy, sure.
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 🙌🏻❤️✨
Totally true. So many resources won't even through a single bone to help intuition. It's definition/proof, barely alluding to novel examples. Throwing in integers mod P in this video really turbo-charged the intuition factor.
Yeay math. Please do videos on topology, real analysis and just any pure math subject you like.
Yes!!! Please do videos on real analysis!
I love Socratica too.. it is everything that a good channel should be.
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. :)
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.
The quality of your teaching is way beyond the average
I thought I found some very good resources over the years, but I am amazed at how I didn't come across Socratica until now. This is the first video of theirs that I have ever seen, and everything from the clear explanation and clean presentation to the really satisfying sound effects is top-notch. I am thinking I may have just started another binge-watch tonight...
you have explained one of the most difficult math topics and made it look easy. I wish you were my prof in University
No this is not one of the most difficult math topics
"Additive inverse" = "opposé" in french
and "multiplicative inverse" is simply "inverse"
Great video and really appreciated work. To provide great video without any cost is a noble work. Be with us and provide more videos on real-analysis :)
Your work I highly valued by myself, I can easily read through a textbook after watching your videos. You are so good!
Just discovered this channel. Instant subscription! I LOVE the style of your exposition!
Thank you Socratica, very cool
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.
Socratica is a companion indeed, you make me feel safe. God bless you, and I hope to be a Patreon soon
I always love the math videos on this channel
A good and fun video that we can watch smiling from beginning to end
Yess!! Socratica We love to watch your videos because these build best concepts...Thank you so much
Thanks for making ideas of fields more clear.
Hope you will make video on Galois fields and their applications.
This is amazing. It took me 30 seconds of watching this video to understand what i have been taking for granted in high school
Great explanation! Covered in less than 10 minutes what I spent an hour searching for. Sub and like 👍🏼
Thanks for the video, pretty straight. The educational approach is awesome, good work !
Videos like these make me fell in love with Mathematics more and more.................. This is the best channel to learn mathematics!!!!!!!
You are the best at explaining these concepts which are somehow complicated. Thanks for making these video
Thanks Socoratica
from Somalia
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 :)
I am amazed by your explanation, it seem much easier now, thanks a lot!
Hey socratica, can you do a series about Galois Theory and Polynomials? since that would be a nice follow up from your abstract algebra series and a nice refresher for the audience who may have done it in the past. Great videos :)
Always great content, well edited. Thank you. Complex number are a pleasure to work with? Since when?
you and your team are so great, i do really appreciate your work! i understand more now , thank you
good comment I like you , i live in India
I was waiting for Field videos when I was taking Abstract Algebra in my junior year. Now, I have even completed my bachelors. Lol
Great video. This is my current course so I greatly appreciate the clarity
Thank you for your kind words! Good luck in your course this term!! 💜🦉
Thanks for uploading these valuable videos. Please also upload videos on functional analysis and complex analysis
Awesome as usual.
Love this! More topology and the like (maybe even do a video on non-orientable surfaces)!
I love the way you explain things...JUST BEAUTIFUL
Wowwww. Just Wowww.
Can't even explain how good it is.
I love these videos. Thank you!
You are doing a great job SOCRATICA...please carry-on...Cover some topics of Differential Geometry if possible...
Just to the point that's what make wonderful lectures ... Thank you Ma'am 😊
wonderful work!!
You guys closed a black hole in my math knowledge, keep up the good work
Thank you so much for this GREAT video!!!
Finally I understood what is a field, thank you!
You explained all of this in best possible way ....you should go more then that would ne reallllly helpful .
It's been a while I've been dreaming of a Socratica-like definition of a Function Field. I wonder if that will come true someday.
Thank you. This video was perfect and helped me a lot.
Great Work🔥
Perfectly explained, thanks
Great Jop 👍👍... Thank You Soooooo Much for these wonderful lectures 🙏🙏🙏
Thanks for such a great explanation
Well explained!
Thank you. That was very clear.
thank you so much. I am studying for a quiz and doing homework and this helped so much
This is the most easy way to understand mathematics you are have a simple and deep understanding of mathematics thanks
Great Video. Thanks for making this.
We look forward to more new videos, please. great contribution.
Thank you so much...this helped me a lot
This channel is great!! Thanks!!
This is very helpful keep up the good work. I will donate when I can.
wish this was there when I was preparing for the exam ! GREAT VIDEO !!!
I just binge watched all of Abstract Algebra. I started trying to makes sense of GCSE math (its unstructured memorization). Between here and numberphile we have what makes sense and interesting.
Thank you this's video very amazing and powerful content.
Beautiful explanation✨
I was able to understand our lesson because of your videos. Next content please about Quasigroup. Thank you in advance!
The best explanation in the internet.
thank you so much! for explaning group/ring/fields.
Thanks! Great explanation of Fields!
Thank you so much for your kind support! It makes a huge difference!! 💜🦉
Very nice video to learn abstract algebra in simple manner with simple english. Excellent work my teachers.... Thank you so much....
keep up the good work! :)
Mind blowing clear definition of field awesome 👌
Just wow...Great explanation
Such a clear explanation even highschooler could understand. Very good, thanks
Got a good revision.thank you
Man I love this Channel
This is awesome!
I mean wow 😲,what an explanation,just amazing❤
YOU ARE AMAZING!!!!
THANK YOU! MORE VIDEOS PLEAASE! :))
Zamn!!! Great explaination
Thank you for the great video. ❤️
Thank you!!!
Thank you so much!!
Great video!
thank you
Well-explained.
You are doing good for the whole mankind. Thank you.
This is really good, thanks
Legendary explanation❤🙏🏻✌🏻
She is a best teacher ..In my thinking ...
Such sweetness in the end can't donate now surely in future 🙂
Great work
Very good explanation. I lost you in what exactly is the Char(F). Maybe it needed a little bit more explanation. Or maybe I should study Galois Theory xD
Char(F) is the smallest number of ones to be added in order for it to be zero. In Z/5Z, 1+1+1+1+1 (5 times) = 0
OMG this was awesome. thank you so much
Excellent!
Great video.👏
I really like your videos (y).
This person is a genius - thx so much