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NOTATIONS
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a^b = a to the power b

sqrt(x) = square-root of x

a*b = product of a and b

a# = a degree
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pi = ||

Euclidean Constructions :A Motivation of Galois Theory

In high-school geometry we have gone through Euclidean Constructions -
namely constructions using straight edge and compass.It goes like this :  we are
given two points (which are assumed to be "constructible")to start with and are
allowed to do the following operations:-

(i)Drawing a circle whose centre is a constructible point and passes through
another constructible point.

(ii)Joining two constructible points by a straight line.

The points of intersection of two circles or two straight lines or a circle and
a straight line obtained by operations (i) and (ii) also become constructible.A
point in the plane is said to be constructible if we can construct it (in the
above sense)by performing operations (i) and (ii) finitely many times.

An obvious question that may arise is :  What are all constructible
points ? Though construction is a geometric phenomenon,to answer this question
we need a bit of Field Theory.In other words,there is something "seriously
algebraic" about Euclidean Construction so that even in order to know how
far-reaching it is we have to use tools from Algebra.

To model Euclidean Construction algebraically let us fix a coordinate
system.Without loss of generality we assume the given points are (0,0) and
(0,1).We say a number x is constructible if the point (0,x) is constructible.
It is a trivial observation that a point is constructible if and only if both of
its coordinates are constructible numbers.So it is enough to find the set of
all constructible numbers.Let us denote this set by C .Then the set of all
constructible points is C X C by our observation.

We can rephrase the definition of a constructible number.A number x is
constructible if and only if we can construct a line-segment with length x.
With this observation we are able to see that sum and difference of two
constructible numbers are again constructible.What about product and ratio ?  In
fact they are also constructible.In order to construct them we have to construct
similar triangles in a clever fashion.We can summarize all this observations by
saying

C is a subfield of R.

Now we understand why it is so important to study field-theory in order
to study Euclidean Constructions.By our choice of coordinate system 1 is in
C.Since C is a field it follows that Q is sitting inside C.Question is :  What
else is in C ?  For example sqrt(2) is in C because by Euclidean Construction we
can construct a line-segment whose length is the square-root of a given
line-segment.So square-root of a constructible number is again constructible.
That means numbers like sqrt(2/3),sqrt(1+2*sqrt(7/sqrt(5+sqrt(6/11)))),
(3-sqrt(52))/sqrt(11) are all constructible.In a word any number that can be
expressed using integers and the symbols +,-,X,/,sqrt are all constructible.
Formally speaking,

If we have a tower of subfields of R such that each of them
is obtained from the previous one by adjoining a square-root,
then the topmost member of the tower is a subfield of C.

Still the question remains :  Is there any other element in C ?  i.e.
is there a constructible number which can not be expressed using integers and
the signs +,-,X,/,sqrt ?  Answer to these questions is NO.  To see this we have
to notice what we are doing at each stage of a Euclidean Construction.Say at
some stage all the points in the picture have coordinates in a field F.(In the
initial stage we take F to be Q .)  When we intersect two straight lines
obtained by joining two pairs of already constructed points,we actually solve a
pair of linear equations with coefficients in F and hence the point of
intersection also has coordinates in F.On the other hand intersecting a
straight line with a circle or two circles amounts to solving a quadratic
equation with coefficients from F.So the points of intersection have
coordinates in F or a quadratic extension of F according as the discriminant is
a square in F or not.Thus at each stage we are at most adjoining a square-root
and stepping up to a quadratic extension.This shows

x is constructible if and only if x belongs to a subfield
of R which can be obtained by finitely many successive
adjunction of square-roots starting from Q.

As a result we see that a constructible number x lies in a field
extension of Q of degree 2^n for some n.That means x is algebraic of degree 2^m
for some m less than or equal to n.This shows for example cube-root of 2 is not
constructible.Geometrically speaking "duplication of a cube" can not be done by
Euclidean Construction.

What about trisecting an angle ?  Can we do it with straight edge and
compass ?In fact we can not trisect all angles by Euclidean Method.For example
take the angle 60#.We will show that 20# can not be constructed.It is quite
clear that an angle A can be constructed if and only if we can construct the
length cos A .Using some trigonometric juggleries we can find that cos(20#)
satisfies the cubic equation 8x^3-6x-1,which happens to be irreducible over
Q.Thus it follows that the degree of cos(20#) over Q is 3 and hence is not
constructible by Euclidean Means.

That 20# can not be constructed can be restated as "a regular 18-gon can
not be constructed".In general we may ask the question :When is a regular n-gon
constructible by Euclidean Method ?  Instead of going into this general question
let's take a concrete example.Take n=11.A regular 11-gon is constructible if and
only if cos(2*pi/11) is constructible.If it is constructible then cos(2*pi/11)
has degree 2^m over Q for some m.As a result a=cos(2*pi/11)+sin(2*pi/11) is
inside a field extension Q(cos(2*pi/11),i) of degree 2^(m+1) over Q.That means
the degree of a over Q is a power of Q.But since 11 is a prime and a is a
primitive 11th root of unity,it follows that the irreducible polynomial of a is
x^10+x^9+...+x+1.Hence regular 11-gon is not constructible.Observe that we have
not used anything but the facts that 11 is a prime and 10 is not a power of 2.So
we have

If p is a prime and (p-1) is not a power of 2 then a
regular p-gon cannot be constructed by Euclidean Means.

But what about the converse ?  i.e.  if p is a prime such that p-1 is a
power of 2 then can we say that a regular p-gon is constructible ?  In fact the
answer is yes.But the proof is not that easy.The reason is simple - earlier when
we were proving a number is not constructible we were calculating its degree
over Q and showing that it is not a power of 2.But when we have a number(x,say)
whose degree is a power of 2 we are not at all in good shape.In order to prove
that x is constructible we have to do one of the following :-

(1)Give an explicit geometric construction of the length x.

(2)Find a tower of subfields of R starting from Q such that each subfield is a
quadratic extension of the previous one and the topmost one contains x.

In general (1) is a difficult job.For example if we need to show that the
regular p-gon is constructible for all p of the form p=2^r+1,we have to
construct the polygon for all such p.This is practically impossible.So we have
to go for (2).But how to do that ?  That is where a systematic study of field
extensions comes into play and this systematic study is done through GALOIS
THEORY.

In Galois Theory we attach a group to each finite field extension.Then
we try to study the bigger field and the intermediate subfields using the
group.As groups are simpler objects compared to fields,this approach works well
provided the field extension is sufficiently well-behaved.

Suppose K be a finite extension of Q.We say f is a Q-automorphism of K
if f:K-K is a field automorphism which keeps all elements of Q fixed.This is
actually a map which preserves all the properties that K has as a field
extension of Q.It is trivial to observe that Q-automorphisms form a group,
called the Galois-group of K/Q.Let us denote it by G(K/Q).This is the group we
were talking about.It can be shown that it is a finite group whose order divides
the degree of K over Q.When equality holds i.e.  when |G(K/Q)|=[K:Q],we say that
the extension K is a Galois extension of Q.Galois extensions enjoy the property
that if a field is Galois over a smaller subfield then it is automatically
Galois over a bigger subfield as well. Best thing about Galois extensions is
the one-to-one correspondence between field extensions of Q sitting
inside K and the subgroups of the Galois group.This correspondence also satisfy
the property that the degree of the intermediate extension equals the index of
the corresponding subgroup in the Galois group.Thus,once we find all the
subgroups of the Galois group, we have all the intermediate field extensions as
well as their degrees.

One way of proving that a number is construtible is to show that it is
inside a field K sitting in-between R and Q in such a fashion that K/Q is Galois
of degree 2^n.This is because the Galois group (whose order is also 2^n) has a
subgroup of index 2.That means we have an intermediate quadratic extension L/Q.
Since K/L is again Galois and of degree 2^(n-1) it follows by induction on n
that K is obtained from Q by successive square-root adjunctions.For any prime
P = 2^r+1,it can be shown that K=Q(cos(2*pi/p)) is a Galois extension of Q of
degree 2^(r-1).Thus using the machineries of Galois Theory we can establish the
constructibility of the regular p-gon quite easily.```