Full transcript
0:05What exactly is a field ? First of all ,
0:08it has a set , which we’ll call K ,
0:10because we’re looking at fields . You
0:12can look up what exactly a set is in
0:14our video on that topic . All you need
0:17to know here is that it contains
0:18objects that you can distinguish from
0:20each other . So , different numbers ,
0:22vectors , ah , whatever you like . Our
0:24favorite also has two operations . The
0:26circle and the square act as
0:28placeholders for them , as always . The
0:31order is really important here , just to
0:33clarify what an operation actually was .
0:35Something like plus , times , minus , and
0:37divided . Well , we won't make it quite
0:39that easy again . Of course , there are a
0:42few more things that must be met , but
0:43to explain those , we need another term
0:45that will be brutally important in your
0:47studies from now on . The term " group , "
0:50because that’s likely how your
0:51professor defines a field , just so you
0:54aren’t confused . We have a separate
0:56video on that , but we'll summarize the
0:58most important things about a group
0:59here again . A group has a set , but only
1:03one operation . For such a set with an
1:06operation to be a group and not just
1:08something meaningless , three things
1:09must be met . First , associativity . A , B
1:14, and C are any elements . No matter
1:16which two you combine first , you get
1:18the same result . So , with an
1:20associative operation , you can just
1:22place parentheses anywhere , but just
1:25don’t swap the order of the letters .
1:27Second , it also needs the existence of
1:30a neutral element . As always , let’s
1:33call the mysterious neutral element E.
1:36The neutral element can be combined
1:38with any other element in any order ,
1:40and the result is always the other
1:42element . The last thing that must be
1:45met is , thirdly , the existence of an
1:47inverse element . For every element A in
1:50the set , there is also an inverse
1:52element a to the power of -1 . Combining
1:55these two elements yields the neutral
1:57element . The order is , once again ,
1:59completely irrelevant . If these three
2:02things are met , the set G with the
2:04operation is called a group . There is
2:07one more thing that is optional , number
2:094 . Fourth , commutativity . With
2:12something commutative , the order in
2:13which you combine the elements simply
2:15doesn't matter . If all four of these
2:18conditions are now met , then the group
2:20is called an Abelian group . Then
2:23we’re done with the recap . Let’s
2:25return to our field . Wait , wrong way ,
2:27over here . We need one more thing now ,
2:30namely something that has nothing
2:32directly to do with groups :
2:33distributivity . For that , we finally
2:36need two operations . Then we need any
2:39three elements from our field set . The
2:42square should now act like
2:43multiplication when expanding terms .
2:45Then the square is called distributive .
2:48When the circle is inside the
2:49parentheses on the right , this is the
2:50result . A square B circle C = A square
2:54B circle A square C. Yes , exactly . Aha .
2:59You can think of it like plus and times
3:01. A * ( B + C ) = A * B + A * C . Sure , if there
3:05is a left-sided one , there is also a
3:07right-sided one . Although in math , you
3:09really never know . In any case , this is
3:11what should come out from the other
3:13side . ( B circle C ) square A = B square
3:15A circle C square A. Because the order
3:18doesn't matter for plus and times , both
3:20sides are naturally the same . So ,
3:23finally . We now have everything we need
3:26. On to the most important question of
3:28your life , at least in the last 5
3:30minutes , if at all . In any case , when
3:32is a set with two operations a field ?
3:35First of all , the set must be an
3:37abelian group with the first operation .
3:40That means associative , commutative ,
3:42with a neutral and an inverse element .
3:45The neutral element of the first
3:47operation is called the zero element .
3:49Why ? Coming up . Second , the set
3:52excluding the zero element must also be
3:55an abelian group with the second
3:56operation . So again , the same four
3:59things here , only that the neutral
4:01element of the second operation is
4:02called the one element . The addition
4:05about excluding the zero element is
4:07damn important , don't forget it . We
4:09have one last condition . The square
4:12must be distributive with the circle .
4:15That is why we explained distributivity
4:17earlier . If all of this holds , then we
4:20have a field . Phew . Oh yes , why the
4:23strange names zero element and one
4:25element ? The standard field that you
4:27all know without knowing it : the real
4:29numbers with plus and times . It really
4:31is a field . Feel free to try it out
4:33with some examples . But since you are
4:35probably just as lazy as we are , you
4:36can also just take our word for it . For
4:38plus , the neutral element is 0 . For
4:40times , the neutral element is 1 . Tada !
4:43That is why they are generally called
4:45zero element and one element . Oh , you
4:47know what ? Write in the comments why
4:48the real numbers with plus and times
4:50form a field . Just list every condition
4:53and provide an example for it . A
4:55complete proof would probably go too
4:56far . Unless you are up for it . Either
4:59way , it is definitely a perfect
5:00exercise to understand this stuff . On
5:02the right here , you can find the
5:03complete playlist on basic algebraic
5:05structures . Until then , share and like
5:07us as always , and big hugs and kisses
5:08and so on , ciao .