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.
Be sure to subscribe so you don't miss new lessons from Socratica:
bit.ly/1ixuu9W
♦♦♦♦♦♦♦♦♦♦
Ways to support our channel:
► Join our Patreon : / socratica
► Make a one-time PayPal donation: www.paypal.me/socratica
► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9
Thank you!
♦♦♦♦♦♦♦♦♦♦
We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
www.jmilne.org/math/CourseNote...
♦♦♦♦♦♦♦♦♦♦
Connect with us!
Facebook: / socraticastudios
Instagram: / socraticastudios
Twitter: / socratica
♦♦♦♦♦♦♦♦♦♦
Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro
Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison
Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison
♦♦♦♦♦♦♦♦♦♦

[@Socratica]

Sign up to our email list to be notified when we release more Abstract Algebra content: snu.socratica.com/abstract-algebra

[@biswajitsuklabaidya780]

Can you please add videos on Linear algebra.

[@AnupKumar-bd9eu]

Moomomomoommoommo

[@cicik57]

must not a field have a propert that x*y means adding x+x+x...x y - times, or it can be different?

[@MrAboy97]

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. 😓😣😢

[@NoActuallyGo-KCUF-Yourself]

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

[@donquic]

+ Groups

[@thecampverdekid806]

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

[@streampunksheep]

Best companion to self learning mathematicians.

[@TheWolfgangGrimmer]

Along with Khan Academy, sure.

[@ishikasharma1000]

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 🙌🏻❤️✨

[@cadaankaa]

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.

[@nic741]

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.

[@michaelnovak9412]

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

[@evelynbrylow3624]

Yes!!! Please do videos on real analysis!

[@stuartneil8682]

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.

[@PunmasterSTP]

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...

[@andreibacila3175]

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

[@alexkorocencev7689]

Thank you Socratica, very cool

[@njabulomahlalela2912]

Your work I highly valued by myself, I can easily read through a textbook after watching your videos. You are so good!

[@erikolsen1333]

I always love the math videos on this channel

[@MatheusSouza-bf9ud]

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

[@webdeveloper-vy7hb]

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 :)

[@krishnasharma-hi5tr]

Videos like these make me fell in love with Mathematics more and more.................. This is the best channel to learn mathematics!!!!!!!

[@maxamedcabdi6397]

Thanks Socoratica from Somalia

[@aishasaddiqa8345]

Yess!! Socratica We love to watch your videos because these build best concepts...Thank you so much

[@Goejii]

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

[@JWentu]

Just discovered this channel. Instant subscription! I LOVE the style of your exposition!

[@rogerwilcoshirley2270]

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.

[@cobbiepeniels6437]

Socratica is a companion indeed, you make me feel safe. God bless you, and I hope to be a Patreon soon

[@golhare]

Thanks for making ideas of fields more clear. Hope you will make video on Galois fields and their applications.

[@spiderjuice9874]

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 :)

[@patrickbyamasu1353]

You are the best at explaining these concepts which are somehow complicated. Thanks for making these video

[@fjanbazi]

You guys closed a black hole in my math knowledge, keep up the good work

[@mathswithmunira8676]

I love the way you explain things...JUST BEAUTIFUL

[@Sam-tb9xu]

Great explanation! Covered in less than 10 minutes what I spent an hour searching for. Sub and like 👍🏼

[@wassimrharbaoui7333]

Thanks for the video, pretty straight. The educational approach is awesome, good work !

[@dinny4001]

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

[@antoniusnies-komponistpian2172]

No this is not one of the most difficult math topics

[@farhanislam8463]

Great Video. Thanks for making this.

[@magdalenatopolewska814]

I am amazed by your explanation, it seem much easier now, thanks a lot!

[@rikenm]

I was waiting for Field videos when I was taking Abstract Algebra in my junior year. Now, I have even completed my bachelors. Lol

[@bckzilla]

Awesome as usual.

[@yunus3561]

Perfectly explained, thanks

[@sebastianvazquez310]

Man I love this Channel

[@skittles6486]

Wowwww. Just Wowww. Can't even explain how good it is.

[@derciferreira7211]

Finally I understood what is a field, thank you!

[@EclipZeMuzik]

wonderful work!!

[@oliviamankowitz8121]

I love these videos. Thank you!

[@gcoronado4622]

Thank you so much for this GREAT video!!!

[@BAstroboy]

Thanks! Great explanation of Fields!

[@Socratica]

Thank you so much for your kind support! It makes a huge difference!! 💜🦉

[@nandha0150]

The best explanation in the internet.

[@joaquin-chw4924]

We look forward to more new videos, please. great contribution.

[@benterrell9139]

Great video. This is my current course so I greatly appreciate the clarity

[@Socratica]

Thank you for your kind words! Good luck in your course this term!! 💜🦉

[@ivideos7348]

Just to the point that's what make wonderful lectures ... Thank you Ma'am 😊

[@navjotsingh2251]

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 :)

[@ghadeeryousif2602]

Beautiful explanation✨

[@chenlightrain]

you and your team are so great, i do really appreciate your work! i understand more now , thank you

[@rajendramaurya9062]

good comment I like you , i live in India

[@chaitrajambigi3426]

Thanks for such a great explanation

[@Konchunas]

Such a clear explanation even highschooler could understand. Very good, thanks

[@abidsyed9534]

Nice description of fields

[@adhinvs7132]

Great Work🔥

[@funwithwaniarumaisaareeba7586]

Thanks for uploading these valuable videos. Please also upload videos on functional analysis and complex analysis

[@joeyquiet4020]

thank you so much! for explaning group/ring/fields.

[@v.gopalakrishnan350]

Well explained!

[@josevitorcavalcante996]

Thank you. This video was perfect and helped me a lot.

[@alexdukhan]

Love this! More topology and the like (maybe even do a video on non-orientable surfaces)!

[@mohit0901]

wish this was there when I was preparing for the exam ! GREAT VIDEO !!!

[@Nipun_Chauhan]

Zamn!!! Great explaination

[@proximacentaur1654]

Thank you. That was very clear.

[@Albert-fe8jx]

Always great content, well edited. Thank you. Complex number are a pleasure to work with? Since when?

[@noraorzack3979]

thank you so much. I am studying for a quiz and doing homework and this helped so much

[@charitylyngdoh8912]

Great Jop 👍👍... Thank You Soooooo Much for these wonderful lectures 🙏🙏🙏

[@adityabodkhe914]

You explained all of this in best possible way ....you should go more then that would ne reallllly helpful .

[@newsgo1876]

You are doing good for the whole mankind. Thank you.

[@waelmarzouk3337]

This is the most easy way to understand mathematics you are have a simple and deep understanding of mathematics thanks

[@ardrak.t3243]

Just wow...Great explanation

[@joydebroy8]

You are doing a great job SOCRATICA...please carry-on...Cover some topics of Differential Geometry if possible...

[@GameChanger-ol1ef]

Great video.👏

[@piyushbansal9716]

Such sweetness in the end can't donate now surely in future 🙂

[@BR-hi6yt]

This person is a genius - thx so much

[@Kishore8219]

Mind blowing clear definition of field awesome 👌

[@tejasgupta1412]

Got a good revision.thank you

[@caiofdacosta]

This channel is great!! Thanks!!

[@lusy3530]

Thank you this's video very amazing and powerful content.

[@aliawde4970]

Legendary explanation❤🙏🏻✌🏻

[@leepatrick1756]

Excellent!

[@gulcedemirel9820]

OMG this was awesome. thank you so much

[@muzafarhussain6878]

She is a best teacher ..In my thinking ...

[@lmsawmi]

Thank you for the great video. ❤️

[@djeovs]

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.

[@louisscott471]

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.

[@dhyangandhi4344]

Thank you so much...this helped me a lot

[@silversky216]

Damn the race was engaging and exciting!!!

[@farhan2373]

Thank you so so much!

[@sujitmohanty1]

No doubt these teachings are class apart!

[@madanravuri]

Very nice video to learn abstract algebra in simple manner with simple english. Excellent work my teachers.... Thank you so much....

[@Ali-wl6pm]

very good video!

[@real_anil9688]

You sure make the mathematics understanding a quite easier

[@anusha1113]

I can't stop falling in love with maths because of ur way of teaching mam

[@aaroncollings9949]

This is very helpful keep up the good work. I will donate when I can.

[@sunnyvishnoi4310]

I mean wow 😲,what an explanation,just amazing❤

[@d-shiri]

Well-explained.

[@fredwooten14]

Great video!

[@user-um7tw6kx4r6]

This is really good, thanks

[@vishalbehera2493]

best explanation for self learners. thank you

[@user-yx5uh8lb8d]

This is awesome!

[@edh615]

Auto-subscribed, don't even need to look at content of the channel, you already deserve it with this video.

[@riariopiero7313]

Great work