What is a field ?

Definition of a Field
In this video, I define the concept of a field, which is basically any set where you can add, subtract, add, and divide things. Then I show some neat properties that have to be true in fields. Enjoy!
What is an Ordered Field: • What is a real number?
Check out my Real Numbers Playlist: • Real Numbers

[@bertrandspuzzle]

Whenever I see the long list of axioms for fields (or vector spaces) I think of how happy it makes me to condense the definition with group theory! "A field is a ring whose nonzero elements form an Abelian group." That is where I like to hide my axioms.

[@user-jc2lz6jb2e]

A field is a commutative ring with only 2 ideals.

[@douglasstrother6584]

You *sneaky* Math Geek! ;)

[@bertrandspuzzle]

@@user-jc2lz6jb2e I felt there was shorten the character count! Commutative -> Abelian didn't save too many characters.

[@paulestrada961]

Thabk you Group Theory. Made Linear Algebra seem much less intimidating.

[@suscriptor01]

You sure love your work, don't you? Great video, as always!

[@sciencewithali4916]

Thanks a lot ! Am inspired a lot in my own video making ! Wish one day to be as comfortable as you are ! Excellente continuation

[@CliffStamp]

Wow, that was a really elegant demonstration of a0=0 from 0+0=0

[@erwinmulder1338]

I remember this from university. The icing on the cake is the proof that is 1 in fact larger than 0. Note how the concepts 0 and 1 have been defined as concepts in addition and multiplication, but we haven't proven anything beyond that.

[@plaustrarius]

The integers under addition and multiplication with a prime modulus, if i remember correctly thats the first field i came across. Finite fields really blew me away, (x+y)^2 finally is just x^2 + y^2 if modulo 2! Fermat's little theorem was hard for me to use until i got a better understanding of finite fields. Thank you Dr. Peyam!

[@liyi-hua2111]

Love your videos. Hungers for more.

[@dgrandlapinblanc]

Ok. Thanks.

[@cobalt3142]

This was a great introduction to the concept of fields! However, I think you forgot the most important rule for fields: 1 =/= 0. We can't allow that pesky 0 ring into this exclusive club, eh?

[@user-jc2lz6jb2e]

Allow 0 = 1 if you wanna have some F_un.

[@aneeshsrinivas9088]

Why not?

[@more_math]

Hi Dr Peyam, thanks for the teaching. I have a question for you. When you said “ when you take two elements in your field and add them it’s still in your field”, can the 2 elements be the same element? For example 2+2? Thanks in advance.

[@mertunsal7335]

Mr. Peyam it is not deeply linked but I wanted to ask you something about vector space axioms. I saw a theorem in slader(which I think is not true): V is a vector space over the field F if and only if for every x,y in V and a,b in F , we have ax+by in V But I think vector space axioms are not equivalent to this theorem since this has nothing to do with commutativity and associativity axioms of vector spaces. Thanks in advance.

[@polodelmar9852]

Keep in mind that the vector is a module in a field :)

[@beardymonger]

Your videos are great, I like them!! You are one of my YT heros, thank you very much! Personally, I prefer a wall of examples and motivations - basically stories, calculations and hand weaving :-D - to a wall of definitions. For this reason I very much like books by H. M. Edwards like "Galois Theory" or "Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory"

[@paulestrada961]

Out of curiosity, do fields have to be defined with the two binary operations which mimic addition and multiplication as seen with the real numbers or can a field be defined with any two arbitrarily binary operations which interact with eachother to satisyfy the distributive properties of a field? Also, are additive and multiplicative inverses the only type of inverses with respect to the elements and binary operations of a potential field we can find? For example, are there Fields where say multiplication is the binary addition and exponentiation is the binary multiplication? The topic of fields has always caught my interest.

[@heeraksharma1224]

Great vid,Dr P

[@iabervon]

I think it would be nice to mention GF(2^8) in passing (without actually constructing it), so that you can point out a number of properties that the fields people are familiar with have that fields don't have to have. Then, when you prove things that are obvious, people can keep in mind that not everything that's obvious is even necessarily true of every field.

[@willnewman9783]

Why 2^8 in particular?

[@greghansen38]

Now I wonder if it's even possible for associativity or distributivity to be false. I've seen non-commutative, but they always associate and distribute.

[@JasonOvalles]

I have a question about the proof for the first theorem at 11:01 Why is it that we can add (-c) on both sides? Maybe it is obvious, but I can't see what in the definition of a field allows us to make this move.

[@drpeyam]

By definition of -c

[@JasonOvalles]

@@drpeyam wow Dr Peyam responded to me! I feel so honored, thank you for taking the time. I should've been more specific with my question. What is it about a field that allows us to add a value to both sides of an equality and maintain equality? That is, given a, b, c in a field and a=b, how can we justify a+c=b+c? Is it something about the definition of equivalence that is just unsaid?

[@drpeyam]

Oh, that’s because of the definition of addition as a function. Namely if f(x,y) = x + y, then you’re saying that if a = b, then f(a,c) = f(b,c) which is true for functions

[@MrBeen992]

THANKS !!!!!

[@sparsetable]

Can you explain why do you always write the interrogation sign (?) preceded by a blank space?

[@sparsetable]

It definetly catch my attention btw.

[@drpeyam]

Oh, I think it’s more searchable that way, but maybe I’m mistaken

[@de29a15]

In French, there is this basic rule where you put a space after "one part punctuation" (.,) and a space before and after two part punctuation !!!

[@alexdemoura9972]

Dear Dr. Peyam, at 9:16 : " {0, 1} is a field as long you define 1 + 1 = 0" ▪︎Is there any preceding video of yours about such _"cheating definitions"_ ? ▪︎Why not consider {-1, 0, 1} as a field example instead? ▪︎ If such _"definitions"_ are possible, why not consider n = n-¹ = 1/n and then make ℤ a field? ▪︎ Isn't such _1 + 1 = 0 definition_ breaking the rule of the *0* Definition and even maybe the *1* Definition? Sorry for this 3-week doubts, but I didn't find myself comfortable after few readings on fields.

[@ajayananddwivedi7154]

This was asked in our mathematics test

[@shandyverdyo7688]

Is field a ring? Are they the same thing?

[@drpeyam]

No in a ring you could have ab = 0 without a or b being 0

[@Apollorion]

How many sporters want to play on such kind of fields?

[@nadiyayasmeen3928]

Press F to pay respects

[@SefJen]

For some authors, there are non-commutative fields (for multiplication).

[@user-jc2lz6jb2e]

This is an old convention, and now non-commutative fields are called skew-fields/division rings. (I know in French they still use the old naming)

[@SefJen]

@@user-jc2lz6jb2e Ah ok I'm French. For a non-commutative field we say "corps gauche ". And "corps commutatif " for a commutative field. ^^

[@paulestrada961]

This reminds me of when I went to a conference and the topic was non-associative groups.... which didn't make sense to me while I was also in an Abstract Algebra class for my first time.

[@cobalius]

A place where your bread grows UwU

[@cactuslover2548]

Ah, I love the smell of freshly harvested bread.

[@aneeshsrinivas9088]

I bought your field axioms tshirt

[@drpeyam]

Thank you!!

[@aneeshsrinivas9088]

Why is the requirement that 0≠1 in (M3) even there? This feels like arbitrary descrimination against the set {0}. Does this ruin any precious theorems about fields or something? Is {0} so ill behaved that we dont consider it a field?

[@drpeyam]

Ok

[@thedoublehelix5661]

A field doesn't have to be albielian right?

[@user-jc2lz6jb2e]

It has to be. Are you French?

[@user-lm7yx7wj5l]

FFF

[@thibautlescure7183]

next video proof that all finite fields are cyclic ? :)

[@pierreabbat6157]

Why can't you grow wheat in Z mod 6? Because it's not a field.

[@drpeyam]

LOL, yes!

[@cerwe8861]

𝕞𝕒

[@patolorde]

F

[@OndrejPopp]

What is a field? Ok! Let me think! .......................... thinking ...................................... still thinking ....................................... Oh yeah! Now I remember : kzbin.info/www/bejne/iGrIcndorL2cY7M

[@thesickbeat]

F