Field Definition (expanded) - Abstract Algebra

[@Socratica]

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[@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

[@jamesheffernan8000]

Agree - Books require memorization of ~250 pages to likely fully understand what was presented here.

[@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.

[@neilclay5835]

The quality of your teaching is way beyond the average

[@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.

[@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

[@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

[@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 :)

[@njabulomahlalela2912]

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

[@fjanbazi]

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

[@krishnasharma-hi5tr]

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

[@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...

[@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 :)

[@MatheusSouza-bf9ud]

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

[@erikolsen1333]

I always love the math videos on this channel

[@cobbiepeniels6437]

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

[@BAstroboy]

Thanks! Great explanation of Fields!

[@Socratica]

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

[@alexkorocencev7689]

Thank you Socratica, very cool

[@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.

[@maxamedcabdi6397]

Thanks Socoratica from Somalia

[@Konchunas]

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

[@anusha1113]

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

[@Goejii]

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

[@waelmarzouk3337]

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

[@aishasaddiqa8345]

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

[@mohit0901]

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

[@patrickbyamasu1353]

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

[@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

[@ChantalsBulgingEyebrow]

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

[@sujitmohanty1]

No doubt these teachings are class apart!

[@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 :)

[@magdalenatopolewska814]

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

[@newsgo1876]

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

[@joaquin-chw4924]

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

[@golhare]

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

[@mathswithmunira8676]

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

[@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!! 💜🦉

[@HXMCPP]

i love her. the only good explanation i found among all the yb bs

[@nandha0150]

The best explanation in the internet.

[@piyushbansal9716]

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

[@cheeneyap3573]

I was able to understand our lesson because of your videos. Next content please about Quasigroup. Thank you in advance!

[@joydebroy8]

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

[@abidsyed9534]

Nice description of fields

[@lusy3530]

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

[@JWentu]

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

[@Sam-tb9xu]

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

[@user_375a82]

This person is a genius - thx so much

[@muzafarhussain6878]

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

[@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

[@adityabodkhe914]

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

[@Kishore8219]

Mind blowing clear definition of field awesome 👌

[@edh615]

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

[@jalisraja1359]

You are the best teacher I have ever come across.

[@aabidmushtaq3243]

Really it is high quality explanation. Watching from Indian occupied Kashmir.

[@real_anil9688]

You sure make the mathematics understanding a quite easier

[@aliawde4970]

Legendary explanation❤🙏🏻✌🏻

[@sebastianvazquez310]

Man I love this Channel

[@wassimrharbaoui7333]

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

[@ivideos7348]

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

[@funwithwaniarumaisaareeba7586]

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

[@vishalbehera2493]

best explanation for self learners. thank you

[@madanravuri]

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

[@ghadeeryousif2602]

Beautiful explanation✨

[@joeyquiet4020]

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

[@farhanislam8463]

Great Video. Thanks for making this.

[@skittles6486]

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

[@nthpowerz.7133]

The beats at 1:56 ! I thought it was my heart thumping really fast because of enlightenment 😂😅

[@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.

[@josevitorcavalcante996]

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

[@adhinvs7132]

Great Work🔥

[@oliviamankowitz8121]

I love these videos. Thank you!

[@kirbymarchbarcena]

She teaches more concisely than my teacher at school

[@homemade_draugr]

Perfectly explained, thanks

[@sunnyvishnoi4310]

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

[@alxjones]

It's worth noting that "division rings" do exist and aren't necessarily fields. As long as the multiplication is noncommutative, it will not be a field. But also commutative rings without multiplicative inverses aren't fields either. So really, they are both the distinguishing features between rings and fields.

[@valeriobertoncello1809]

Cool! For example?

[@paulfoss5385]

@@valeriobertoncello1809 Quaternions.

[@charitylyngdoh8912]

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

[@silversky216]

Damn the race was engaging and exciting!!!

[@derciferreira7211]

Finally I understood what is a field, thank you!

[@aaroncollings9949]

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

[@bckzilla]

Awesome as usual.

[@Albert-fe8jx]

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

[@osolomero9600]

Ojala pronto vuelva Socrática en Español . Felicitaciones por sus videos

[@alexdukhan]

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

[@Ali-wl6pm]

very good video!

[@user-um7tw6kx4r6]

This is really good, thanks

[@taggebagge]

Hello, I think I have understood the concepts of a field quite well overall but I have a question. In my textbook I have a summary that states: I: Any finite field has prime order q = p^r, that is the order of the field |F| is the exponent of a prime factor, 9 = 3^2 etc. II: Additative group is isomorphic to (Cp)^r III: Multiplicative group is isomorphic to (C[q-1]), where we remove 0. My question pertains to I) and III): What happens if we look at a field spanned by 2 or greater exponent, such as 9 = 3^2? That is |F| = 9. If we have Z9 = [0, 1, 2, 3, 4, 5, 6, 7, 8] then we should get that (F/{0},*) = [1, 2, 3, 4, 5, 6, 7, 8] (Since we have to remove 0 for inverse-reasons). But if within (F/{0},*) we pick two elements like 3*3, we should get 0, since 3*3 mod 9 = 0, and that violates the clossure axiom. I understand fully that Z5 is a field since Z5 gives F/{0} = [1, 2, 3, 4] and there are no elements within this range that can ever produce 5 through a binary operation. But for Z9 we have the elements 3*3 = 9 = 0 and that violates clossure? And if we include 0 we have element without inverse (violates field requirement). In summary: I don't understand how we can ever have an exponent larger than 1 for our prime number P spanning the field. Can't we just then take P*P....*P, where P exists as an element, and as such there can be no clossure?

[@taggebagge]

I think I have found an answer to question above: I would think that the solution to what I am wondering lies in the representation of F9 and in representing Fields as polynomials. That is if we have a repeating prime factor, for each repeation we add one degree, and also we remove the 0 (for inverse reasons). So if we have Z9 we have two repeating primefactors 3^2, and we can represent the field as (F/{0}, *) = [1, 2, x, x+1, x+2, 2x, 2x+1, 2x+2] mod 3. We have p = 3 and r = 2 -> polynomial of (degree r-1 = 2-1) spanned by (mod p = 3) creates the field I guess. In this way we can have clossure and remove 0, and at the same time we have this beautiful connection between fields and polynomials that we know exists. So yes, I think the answer to my question is that we represent it as a polynomial and for each exponent we get a new degree of representations. The smart, dark-haired woman in blue in the video used Z5 as an example. Z5 has a primefactor of only 5 so yes, it would be a polynomial but with the degree 1-1 = 0, so it would make sense for it to represent it as [1, 2, 3, 4] mod 5 since all the elements are coprime with 5, there can be no combination of a,b € F so that a*b is congruent to 0. If I am completely off track and deluded, to whoever reads this you may feel ever so free in correcting me. I have a pending exam so any correction as to my reasoning would be greated with 100% gratitude. Still, I think we are on the right track here :) Fantastic video. Mathmatics rocks!

[@v.gopalakrishnan350]

Well explained!

[@ardrak.t3243]

Just wow...Great explanation

[@PETAJOULE543]

Great explanation about fields

[@PETAJOULE543]

Basically field consist of rational numbers and modulo prime integers

[@EclipZeMuzik]

wonderful work!!

[@lmsawmi]

Thank you for the great video. ❤️

[@ShaolinMonkster]

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

[@mzg147]

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

[@gulcedemirel9820]

OMG this was awesome. thank you so much

[@thecampverdekid806]

I hated abstract when i took it, but it helped me understand mathematics more than any other class I took in undergrad

[@mjjeon2292]

thank you so very much for the quality video

[@gcoronado4622]

Thank you so much for this GREAT video!!!