loosely speaking a field is a set of
elements where you can freely add
subtract multiply and divide and where
the usual rules of arithmetic apply
since subtraction is the same as adding
a negative and dividing is the same as
multiplying by reciprocals you'll often
hear the definition of a field as being
a set of elements with two operations
addition and multiplication and that
both operations are connected by the
distributive property all the elements
form a commutative group under addition
and if you throw out zero the remaining
elements form a commutative group under
multiplication this is the more formal
definition let's now look at some
examples of
fields the integers are not a field we
denote the integers by a double stroke z
also known as a blackboard bold z we
have two operations addition and
multiplication the distributive property
holds and under addition the integers
even form a commutative group also known
as an abelian group but we run into
trouble when we look at multiplication
the integers don't have inverses under
multiplication the multiplicative
inverse of two is one-half and one-half
is not an integer the multiplicative
inverse of three is a third also not an
integer and so on
but if we add all the fractions to the
set of integers we get a field the
rational numbers we call fractions
rational numbers because they're all
ratios of integers the set of all
fractions is denoted by a q for quotient
the additive inverse of a fraction say
for example three-fifths is negative
three-fifths which is another fraction
the multiplicative inverse is five
thirds again a fraction so after you
toss out zero the set of fractions all
have multiplicative inverses the
rational numbers are a field
we can take the set of rational numbers
and turn them into a larger field take
for example the square root of two we've
known for thousands of years that this
is not a rational number
is there a field that contains this
number indeed there is
what if we took the square root of two
and tossed it into the set of rational
numbers we then add subtract multiply
and divide all these numbers together
and we do this over and over until we
have a new larger set of numbers which
is a field containing the square root of
2.
here is a notation for this field
we call it an extension field of the
rational numbers a very good exercise is
to convince yourself that every number
in this field can be written in this
form
our previous example opened up a whole
new world of fields for us you could
extend the rational numbers by adding
the cube root of 3 or i the square root
of negative 1.
there are an infinite number of
extension fields of the rational numbers
in fact for any polynomial equation you
can take a solution to this equation add
that to the rational numbers and get an
extension field we call this an
algebraic extension and you study these
a lot in number theory
in calculus you learn about convergent
sequences a series of numbers which get
closer and closer to some value we call
this value the limit of the sequence
this idea will lead us to another field
if you start off with the rational
numbers and then look at all the
convergent sequences
take their limits and throw them into
the set of rational numbers you will end
up with the real numbers a very large
and well-known field
the notation for the real numbers is an
r r for real and because we've added all
the limits and filled in all the gaps we
say the real numbers are complete
there are still some numbers missing
from the field of real numbers i for
example the square root of negative 1.
if you add i to the real numbers then
add subtract multiply and divide
everything repeatedly you end up with
the complex numbers we denote the
complex numbers with the c
you may be thinking at this point can we
go bigger can we make bigger fields by
finding new numbers either by taking
limits of sequences or solutions to
polynomial equations the answer is no
the field of complex numbers is quite
large any sequence of complex numbers
which converges will converge to another
complex number it is complete and any
polynomial equation can be solved
completely using complex numbers we say
the complex numbers are algebraically
closed
so is this the end of the line
can we not make any more fields
actually we can but we have to think a
bit differently what if instead of
adding a new number we added a variable
x
once we add x to the mix we get the set
of rational functions fractions where
the numerator and denominator are both
polynomials believe it or not this is a
field too the field of rational
functions over the complex numbers we
can even make this larger by adding a
second variable y or a third z
i think we should stop there for now as
you can see there are a lot of different
fields in this world
one of the goals in abstract algebra is
to classify all these fields and
investigate their properties
and while we focus today on infinite
fields there are even fields with only a
finite number of elements
these are called finite fields and we'll
talk about those in a separate video
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