NOTATIONS --------- a^b = a to the power b sqrt(x) = square-root of x a*b = product of a and b a# = a degree __ 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.