## Number Theory for dummies

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Number theory for dummies

Euler's interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. A lot of Euler's early work on number theory was based on the works of Pierre de Fermat. Euler developed some of Fermat's ideas, and disproved some of his conjectures. Euler linked the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers; this is known as the Euler product formula for the Riemann zeta function. Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and he made distinct contributions to Lagrange's four-square theorem. He also invented the totient function φ(n) which is the number of positive integers less than or equal to the integer n that are coprime to n. Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss. By 1772 Euler had proved that 231 − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867.

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study
of the integers. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number
theory is the queen of mathematics."[1] Number theorists study prime numbers as well as the properties of objects made out of
integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers).

Integers can be considered either in themselves or as solutions to equations (Diophantine geometry).
Questions in number theory are often best understood through the study of analytical objects
(for example, the Riemann zeta function) that encode properties of the integers, primes or other
number-theoretic objects in some fashion (analytic number theory). One may also study real numbers
in relation to rational numbers, for example, as approximated by the latter (Diophantine approximation).

The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory".
(The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings
in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term
arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence.
In particular, arithmetical is preferred as an adjective to number-theoretic.

Eusebius, PE X, chapter 4 mentions of Pythagoras:

"In fact the said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and Egypt,
and all Persia, being instructed by the Magi and the priests: and in addition to these he is related to have
studied under the Brahmans (these are Indian philosophers); and from some he gathered astrology, from others geometry,
and arithmetic and music from others, and different things from different nations, and only from the wise men of Greece
did he get nothing, wedded as they were to a poverty and dearth of wisdom: so on the contrary he himself became the author of
instruction to the Greeks in the learning which he had procured from abroad."

Pierre de Fermat (1607–1665) never published his writings; in particular, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes.[36] He wrote down nearly no proofs in number theory; he had no models in the area.
One of Fermat's first interests was perfect numbers (which appear in Euclid, Elements IX) and amicable numbers; these topics
led him to work on integer divisors, which were from the beginning among the subjects of the correspondence (1636 onwards)
that put him in
touch with the mathematical community of the day.

In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these integers are further restricted to prime numbers, the process is called prime factorization.
When the numbers are sufficiently large, no efficient, non-quantum integer factorization algorithm is known. An effort by several
researchers, concluded in 2009, to factor a 232-digit number (RSA-768) utilizing hundreds of machines took two years and the
researchers estimated that a 1024-bit RSA modulus would take about a thousand times as long.
However, it has not been proven that no efficient algorithm exists. The presumed difficulty of this problem is
at the heart of widely used algorithms in cryptography such as RSA. Many areas of mathematics and computer science
have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing.
Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime numbers. When they are both large, for instance more than two thousand bits long, randomly chosen, and about the same size (but not too close, e.g., to avoid efficient factorization by Fermat's factorization method), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the primes being factored increases, the number of operations required to perform the factorization on any computer increases drastically.
Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, the RSA problem. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure.

## Least Common multiple

A number g is a least common multiple of all numbers a, b, c, d ,... if and only if the two following conditions hold:1. a, b, c, and d divides g

2. if a, b, c, and d divides a number f then g divides f.

Suppose 3, 4, 5 are three numbers. The least common multiple is 60 because all the numbers divides 60. All the numbers divides 120 also. So 60 divides 120. The two conditions mentioned above are full-filled. So 60 must be the least multiple of 3, 4, 5 .

Integers play a vital role in number theory. Gauss knew about that and warned us thereby.

The classification of real numbers can be represented by a diagram:

## Unique Factorization theorem

Unique factorization theorem is related to some integer k and primes numbers. Here is what the theorem says:Example :

## Constructible numbers

A real number is said to be constructible if , given a line segment of unit length, it is possible to construct from it a line segment of length r using only straightedge and compass. For example number 2 is constructible. Given any line segment AB of unit length, draw a circle centred about the point B to intersect the line at a new point C. Then the length AC is two (2).## Different sizes of infinity

Cantor has created a paradise from where no one can expell us - as the poverb goes. But what did Cantor invent ?The Wrst key ingredient in Cantor’s revolution is the idea of a one-to-one 1–1 correspondence.4 We say that two sets have the same cardinality (which means, in ordinary language, that they have the ‘same number of elements’) if it is possible to set up a correspondence between the elements of one set and the elements of the other set, one to one, so that there are no elements of either set that fail to take part in the correspondence. It is clear that this procedure gives the right answer (‘same number of elements’) for Wnite sets (i.e. sets with a finite number 1, 2, 3, 4, . . . of members, or even 0 elements, where in that case we require the correspondence to be vacuous). But in the case of inWnite sets, there is a novel feature (already noticed, by 1638, by the great physicist and astronomer Galileo Galilei) that an infinite set has the same cardinality as some of its proper subsets (where ‘proper’ means other than the whole set).

Let us see this in the case of the set N of natural numbers:

N = {0, 1, 2, 3, 4, 5, . . . }:

If we remove 0 from this set, 6 we find a new set N- 0 which clearly has the same cardinality as N, because we can set up the 1–1 correspondence in which the element r in N is made to correspond with the element r þ 1 in N - 0. Alternatively, we can take Galileo’s example, and see that the set of square numbers {0, 1, 4, 9, 16, 25, . . . } must also have the same cardinality as N, despite the fact that, in a well-deWned sense, the square numbers constitute a vanishingly small proportion of the natural numbers as a whole. We can also see that the cardinality of the set Z of all the integers is again of this same cardinality. This can be seen if we consider the ordering of Z given by

{0, 1, - 1, 2, - 2, 3, - 3, 4, - 4, . . . },

which we can simply pair off with the elements {0, 1, 2, 3, 4, 5, 6, 7, 8, . . . } of the set N. More striking is the fact that the cardinality of the rational numbers is again the same as the cardinality of N. There are many ways of seeing this directly, but rather than demonstrating this in detail here, let us see how this particular example falls into the general framework of Cantor’s wonderful theory of infinite cardinal numbers. First, what is a cardinal number? Basically, it is the ‘number’ of elements in some set, where we regard two sets as having the ‘same number of elements’ if and only if they can be put into 1–1 correspondence with each other. We could try to be more precise by using the ‘equivalence class’ idea (employed above to define Fp for a prime p; see also the Preface) and say that the cardinal number a of some set A is the equivalence class of all sets with the same cardinality as A. In fact the logician Gottlob Frege tried to do just this in 1884, but it turns out that there are fundamental diYculties with open-ended concepts like ‘all sets’, since serious contradictions can arise with them. In order to avoid

such contradictions, it seems to be necessary to put some restriction on the size of the ‘universe of possible sets’. I shall have some remarks to make about this disturbing issue shortly. For the moment, let us evade it by taking refuge in a position that I have been taking before (as referred to in the Preface, in relation to the ‘equivalence class’ definition of the rational numbers). We take the cardinals as simply being mathematical entities (inhabitants of Plato’s world!) which can be abstracted from the notion of 1–1 equivalence between sets. We allow ourselves to say that the set A ‘has cardinality a’, or that it ‘has a elements’, provided that we are consistent and say that the set B also ‘has cardinality a’, or that it ‘has a elements’, if and only if A and B can be put into 1–1 correspondence. Notice that the natural numbers can all be thought of as cardinal numbers in this sense—and this is a good deal closer to the intuitive notion of what a natural number ‘is’ than the ‘ordinal’ definition (0 = {}, 1 = {0}, 2 = {0, {0} }, 3 = {0, {0}, {0, {0}}}, . . . ) . The natural numbers are in fact the finite cardinals (in the sense that the infinite cardinals are the cardinalities of those sets, like N above, which contain proper subsets of the same cardinality).

Next, we can set up relationships between cardinal numbers. We say that the cardinal α is less than or equal to the cardinal β, and write

α <= β

(or equivalently β => α), if the elements of a set A with cardinality a can be put into 1–1 correspondence with the elements of some subset (not necessarily a proper subset) of the elements of some set B with cardinality β. It should be clear that if α <= &beta and β <= γ, then α <= γ. One of the beautiful results of the theory of cardinal numbers is that, if α <= β and α => β,

then α = β

meaning that there is a 1–1 correspondence between A and B. We may ask whether there are pairs of cardinals a and b for which neither of the relations α <= β and α => β holds. Such cardinals would be noncomparable. In fact, it follows from the assumption known as the axiom of choice that non-comparable cardinals do not exist.

The axiom of choice states that if we have a set A, all of whose members are non-empty sets, then there exists a set B which contains exactly one element from each of the sets belonging to A. It would appear, at Wrst, that the axiom of choice is merely asserting something absolutely obvious! However, it is not altogether uncontroversial that the axiom of choice should be accepted as something that is universally valid. My own position is to be cautious about it. The trouble with this axiom is that it is a pure ‘existence’ assertion, without any hint of a rule whereby the set B might be specified. In fact, it has a number of alarming consequences. One of these is the Banach–Tarski theorem,7 one version of which says that the ordinary unit sphere in Euclidean 3-space can be cut into five pieces with the property that, simply by Euclidean motions A

(i.e. translations and rotations), these pieces can be reassembled to make two complete unit spheres! The ‘pieces’, of course, are not solid bodies, but intricate assemblages of points, and are deWned in a very non-constructive way, being asserted to ‘exist’ only by use of the axiom of choice. Let me now list, without proof, a few very basic properties of cardinal numbers. First, the symbol <= gives the normal meaning when applied to the natural numbers (the Wnite cardinals). Moreover, any natural number is less than or equal to (<=) any infinite cardinal number— and, of course, it is strictly smaller, i.e. less than (<) and not equal to it. Now suppose that β # α, with a infinite, then (in stark contrast with what we are familiar with for Wnite numbers) the cardinality of the union A U B is simply the greater of the two, namely a, and the cardinality of the product AXB is also a. (We have seen examples of the product before. The set AXB is consists of all pairs (a, b), where a is taken from A and b from B. For finite sets, the cardinality of their product, as sets, is the ordinary numerical product of their cardinalities, which for Wnite sets with more than one member is always larger than the cardinality of either individually.) This does not seem to get us very far if we want to Wnd infinities that are bigger than the ones that we have already. We seem to have got 'stuck' at a

We shall be seeing how to get ‘unstuck’ in the next section. For the moment, however, we can see that what we have done above is at least enough to show us that the number of rational numbers is the same as the number of natural numbers. Following Cantor, let us use the symbol ℵ0 (‘aleph nought’ or, in the US, ‘aleph null’) for the cardinality of the natural numbers N which, as we have seen above, is the same as the cardinality of the integers Z. In fact, the infinite number ℵ0 is the smallest of the inWnite cardinals. Now, what is the cardinality &ro; of the rationals? Any rational number can be written (in many ways) in the form a/b, where a and b are integers. Choosing one of these ways (say, 'lowest terms') for each rational, we have found a 1–1 correspondence between the set of rationals with a subset of the set NXN. Therefore &ro; is less than or equal to the cardinality of NXN. But by the above (or, by direct application of Exercise , the cardinality of NXN is equal to the cardinality of N, namely ℵ0. Thus, &ro; <= ℵ0. But the integers are contained in the rationals, so ℵ0 <= &ro;. Hence, &ro; = ℵ0.

## Turing Machines revisited

First, we need a notion of what it means to 'construct' something in mathematics. It is best that we restrict attention to subsets of the set N of natural numbers, at least for our primitive considerations here. We may ask which such subsets are defined ‘constructively’? It is fortunate that we have at our disposal a wonderful notion, introduced by various logicians12 of the Wrst third of the 20th century and put on a clear footing by Alan Turing in 1936. This is the notion of computability; and since electronic computers have become so familiar to us now, it will probably suYce for me to refer to the actions of these physical devices rather than give the relevant ideas in terms of some precise mathematical formulation. Roughly speaking, a computation (or algorithm) is what an idealized computer would perform, where ‘idealized’ means that it can go on for an indeWnite length of time without ‘wearing out’, that it never makes mistakes, and that it has an unlimited storage space. Mathematically, such an entity is eVectively what is called a Turing machine.Any particular Turing machine T corresponds to some specific computation that can be performed on natural numbers. The action of T on the particular natural number n is written T(n), and we normally take this action to yield some (other) natural number m:

T(n) = m;

Now, a Turing machine might have the property that it gets 'stuck' (or 'goes into a loop') because the computation that it is performing never terminates. I shall say that a Turing machine is faulty if it fails to terminate when applied to some natural number n. I call it eVective if, on the other hand, it always does terminate, whatever number it is presented with. An example of a non-terminating (faulty) Turing machine T would be the one that, when presented with n, tries to Wnd the smallest natural number that is not the sum of n square numbers (0^2 = 0 included). We find T(0) = 1, T(1) = 2, T(2) = 3, T(3) = 7 (the meaning of these equations being exemplified by the last one: ‘7 is the smallest number that is not the sum of 3 squares’), but when T is applied to 4, it goes on computing forever, trying to Wnd a number that is not the sum of four squares. The cause of this particular machine’s hang-up is a famous theorem due to the great 18th century French–Italian mathematician Joseph C. Lagrange, who was able to prove that in fact every natural number is the sum of four square numbers. (Lagrange will have a very considerable importance for us in a different context later, most particularly in Chapters 20 and 26, as we shall see!)

Each separate Turing machine (whether faulty or effective) has a certain ‘table of instructions’ that characterizes the particular algorithm that this particular Turing machine performs. Such a table of instructions can be completely specified by some ‘code’, which we can write out as a sequence of digits. We can then re-interpret this sequence as a natural number t; thus t codifies the ‘program’ that enables the machine to carry out its particular algorithm. The Turing machine that is thereby encoded by the natural number t will be denoted by Tt. The coding may not work for all natural numbers t, but if it does not, for some reason, then we can refer to Tt as being ‘faulty’, in addition to those cases just considered where the machine fails to stop when applied to some n. The only effective Turing machines Tt are those which provide an answer, after a finite time, when applied to any individual n.

One of Turing’s fundamental achievements was to realize that it is possible to specify a single Turing machine, called a universal Turing machine U, which can imitate the action of any Turing machine whatever. All that is needed is for U to act Wrst on the natural number t, specifying the particular Turing machine Tt that is to be mimicked, after which U acts upon the number n, so that it can proceed to evaluate Tt(n). (Modern general-purpose computers are, in essence, just universal Turing machines.) I shall write this combined action U(t, n), so that U(t, n) = Tt(n)

We should bear in mind, however, that Turing machines, as defined here, are supposed to act only on a single natural number, rather than a pair, such as (t, n). But it is not hard to encode a pair of natural numbers as a single natural number, as we have seen earlier . The machine U will itself be defined by some natural number, say u, so we have

U = Tu

How can we tell whether a Turing machine is effective or faulty? Can we Wnd some algorithm for making this decision? It was one of Turing’s important achievements to show that the answer to this question is in fact ‘no’! The proof is an application of Cantor’s diagonal slash. We shall consider the set N, as before, but now instead of considering all subsets of N, we consider just those subsets for which it is a computational matter to decide whether or not an element is in the set. (These cannot be all the subsets of N because the number of different computations is only ℵ0, whereas the number of all subsets of N is C.) Such computationally defined sets are called recursive. In fact any recursive subset of N is deWned by the output of an effective Turing machine T, of the particular kind that it only outputs 0 or 1. If T(n) = 1, then n is a member of the recursive set defined by T (‘in’), whereas if T(n) = 0, then n is not a member ('out'). We now apply the Cantor argument just as before, but now just to recursive subsets of N. The argument immediately tells us that the set of natural numbers t for which Tt is eVective cannot be recursive. There is no algorithm, applicable to any given Turing machine T, for telling us whether or not T is faulty!

It is worth while looking at this reasoning a little more closely. What the Turing/Cantor argument really shows is that the set of t for which Tt is effective is not even recursively enumerable. What is a recursively enumerable subset of N? It is a set of natural numbers for which there is an effective Turing machine T which eventually generates each member (possibly more than once) of this set when applied to 0, 1, 2, 3, 4, . . . successively. (That is, m is a member of the set if and only if m = T(n) for some natural number n.) A subset S of N is recursive if and only if it is recursively enumerable and its complement N - S is also recursively enumerable. The supposed 1–1 correspondence with which the Turing/ Cantor argument derives a contradiction is a recursive enumeration of the effective Turing machines. A little consideration tells us that what we have learnt is that there is no general algorithm for telling us when a Turing machine action Tt(n) will fail to stop

What this ultimately tells us is that despite the hopes that one might have had for a position of ‘extreme conservatism’, in which the only acceptable sets would be ones—the recursive sets—whose membership is determined by clear-cut computational rules, this viewpoint immediately drives us into having to consider sets that are non-recursive. The viewpoint even encounters the fundamental difficulty that there is no computational way of generally deciding whether or not two recursive sets are the same or different sets, if they are deWned by two different effective Turing machines Tt and Ts! Moreover, this kind of problem is encountered again and again at different levels, when we try to restrict our notion of ‘set’ by too conservative a point of view. We are always driven to consider classes that do not belong to our previously allowed family of sets.

These issues are closely related to the famous theorem of Kurt Go¨ del. He was concerned with the question of the methods of proof that are available to mathematicians. At around the turn of the 20th century, and for a good many years afterwards, mathematicians had attempted to avoid the paradoxes (such as the Russell paradox) that arose from an excessively liberal use of the theory of sets, by introducing the idea of a mathematical formal system, according to which there was to be laid down a collection of absolutely clear-cut rules as to what lines of reasoning are to count as a mathematical proof. What Go¨ del showed was that this programme will not work. In effect, he demonstrated that, if we are prepared to accept that the rules of some such formal system F are to be trusted as giving us only mathematically correct conclusions, then we must also accept, as correct, a certain clear-cut mathematical statement G(F), while concluding that G(F) is not provable by the methods of F alone. Thus, Go¨ del shows us how to transcend any F that we are prepared to trust

There is a common misconception that Go¨del's theorem tells us that there are 'unprovable mathematical propositions', and that this implies that there are regions of the ‘Platonic world’ of mathematical truths that are in principle inaccessible to us. This is very far from the conclusion that we should be drawing from Go¨ del’s theorem. What Go¨ del actually tells us is that whatever rules of proof we have laid down beforehand, if we already accept that those rules are trustworthy (i.e. that they do not allow us to derive falsehoods), then we are provided with a new means of access to certain mathematical truths that those particular rules are not powerful enough to derive.

Go¨ del’s result follows directly from Turing's (although historically things were the other way around). How does this work? The point about a formal system is that no further mathematical judgements are needed in order to check whether the rules of F have been correctly applied. It has to be an entirely computational matter to decide the correctness of a mathematical proof according to F. We Wnd that, for any F, the set of mathematical theorems that can be proved using its rules is necessarily recursively enumerable

Now, some well-known mathematical statements can be phrased in the form ‘such-and-such Turing machine action does not terminate’. We have already seen one example, namely Lagrange’s theorem that every natural number is the sum of four squares. Another even more famous example is ‘Fermat’s last theorem’, proved at the end of the 20th century by Andrew Wiles ( Yet another (but unresolved) is the well-known ‘Goldbach conjecture’ that every even number greater than 2 is the sum of two primes. Statements of this nature are known to mathematical logicians as π1-sentences. Now it follows immediately from Turing’s argument above that the family of true π1-sentences constitutes a non-recursively enumerable set (i.e. one that is not recursively enumerable). Hence there are true π1-sentences that cannot be obtained from the rules of F (where we assume that F is trustworthy) This is the basic form of Go¨del’s theorem. In fact, by examining the details of this a little more closely, we can reWne the argument so as to obtain the version of it stated above, and obtain a specific P1-sentence G(F) which, if we believe F to yield only true π1-sentences, must escape the net cast by F despite the remarkable fact that we must conclude that G(F) is also a true π1-sentence!

## Galios theory

Galios theory is the connection between field theory and group theory. Galios first used it to study the roots of polynomial equation. It characterizes polynomial equations that are solvable using radicals in terms of the permutation group of their roots. An equation is solvable by radicals if its root can be expressed by formula involving integers, nth roots and arithmetic operations.permutation group is a group whose elements are permutations of some set M and whose group operation is the composition of permutations in G (bijective functions from the set M to itself).

To see the essence of Galios theory we need to know the fudamental theorem of Galios which relates the field of roots of a polynomial to the Galios group( which shuffles the objects ; roots between themselves). So we can gather field structure from group structure and vice-versa. This is known as Galios correspondence. Schematically it can be represented as :

Every field extension has a corresponding group extension. The group elements are the roots of the corresponding field extension. At the top of the tower there is trivial subgroup of { 1} .

Now field is an algebraic object which has certain structure defined by some rules like addition and multiplication(X). Consider an irreducible polynomial P(x) = x^2 + 1 with coefficient in Q ( rationals). One

In the above example Q(i) is an extension of Q. Any field K which contains smaller field K is the extension field of K. The field F which contains all the roots of polynomial f(x) is called the splitting field of f(x).

Now Galios enters the picture. Galios group associated with a nth degree polynomial f(x) C F[x] has its own action. In particular elements of Galios group G is the isomorphism σ : K - K . This field automorphism is the isomorphism of a field to itself. Galios group is isomorphic to a subgroup called permutation group described before.

Since the purpose of permutation group is to permute elements it not hard to see that Galios group G acts on the splitting field F by permuting roots of f(x). If f(x) can be expressed as a product of irreducibe factors like f(x) = f1(x)f2(x)f3(x).... then G will permute roots of f1(x) among themselves , roots of f2(x) among themselves, in other words G acts transitively on the irreducible factors of f(x).

Galios's insight was to see if there were subgroups (H1, H2, .... Hn ) of G such that H(i+1)/H(i) is albenian ( see group theory) then it is possible to write down the formula for the roots of f(x). In this case we say that G is solvable. If not, not.

The necessary condition for solvability of a group G is

In Galio's theory symmetry group plays a crucial role. A subgroup is a smaller of group of elements of a larger group. For example permutations of 1, 2, 3,.. 10 and 2, 4, 6, 8 , 10 form two group G and H. In this case H is a subgroup of G under the action of permutation. A group always involves an action. A permutaion is represented as the two row matrix. We know 1, 2, 3 form six different permutations (3! = 6). Now a permutation on this numbers is represented as a two row matrix.

For example these two matrices below represents two different functions.

function f permutes 1 , 3 and 4, 5. On the other hand g permutes 1,2, 5 and 3,4. Both the action of f and g leaves the group unchanged. And the composition action fog also leaves the group unchanged. The symmetry group is defined over n symbols consisting of permutation operations which can be performed on those n symbols.

Radical is any number that includes the root sign. it is the opposite of exponential.

To understand properly Galios's theory we first need to know what it means to have a solution in terms of radicals and addition (+), substraction(-) , multipliccation and division on them. We can start with quadratic equaion. A quadation equation of the form ax^2 + bx + c = 0 has two solutions of the form

The quantity under the square root is called the discriminant(&del;). There is some kind of symmetry in the roots. For example if 1+√6 and 1-√6 be two roots then we can easily interchange them to find the exact equation of which they are the roots. In case with Galios theory we call this a permutation.

Solving third degree polynomial is also possible using rational expression of finite order. Roots of polynomials are sometimes called zeros. They are the x-intercepts of the graph of the polynomial.

The analogous formula for equation ax^3 + bx^2 + cx + d = 0

A general formula was found by Cardano in 1545 , which has the expression :

There is such closed form expression in terms of radical also for fourth degree polynomials. But these are harder to solve. Fourth degree or quartic equation also has discriminant Δ. A general fourth degree equation of the form ax^4 + bx^3 + cx^2 + dx + e = 0 has a discriminant given by

We define D = Δ^2 and the nature of the roots is determined by D and Δ. If Δ< 0 , equation has two distinct real roots and two are complex conjugate non-real roots. If Δ > 0 all four roots are real and distinct. In general the four roots are expressed by

Where p and q are the second and first degree coefficient of associated depressed polynomial;

Where Δ(o) = c^2 - 3bd + 12ae; and Δ(1) = 2c^3 - 9bcd + 2b^2e + 27ad^2 - 27ace.

Associated depressed polynomial is found by substituting x = y - b/4 . After regrouping the terms we get the desired equation

y^4 + py^2 + qy + r = 0; where

Discriminant of a equation of any degree is defined in terms of roots and coefficients as

Where r1, r2,r3 are the roots of the polynomial. The discriminant of a polynomial can be determined by the qoutient of the resultant of A and A` by a^n

Where A` is the derivative polynomial of A. The resultantant matrix is

where a(n) and b(n) are the coefficients of the corresponding matrices.

## Resolvent of polynomian

Resolvent is the depressed polynomial that has already been discussed. Now we see how it can be used to solve a third degree polynomial. Let a third degree polynomial is--- (1)

Now complete z^3 + 6z^2 to the full cube of (z+2)^3 = z^3 + 6z^2 + 12z + 8 where symbol is the exponential. Now we can rewrite the equation number 1 as w^3 - 3w + 1 = 0 where w = z + 2

Now let w = u + v and make a comparison of this eqation with the form w^3 + pw + q = 0 . We find that uv = 1 and u^3 + v^3 = -1 ; u^3 and v^3 are, thus, the roots of the equation t^2 + t + 1 = 0 . Solving this equation we get

--- (1)

Choosing the cube roots of these solutions to be u = e^(2πi/9) and v = e^(-2πi/9), we obtain

--- (1)

so the final solutions are

--- (1)

There is a general relationship between coefficients and roots of polynomial

And the product of the roots are

As an example for the qubic equation

--- (1)

So if all the roors are given then you can reconstruct any polynomial using the above relationships.

--- (1)

## Lagrange resolvent

Lagrange resolvent is a polynomial that can be analysed to solve another polynomial. There are usually three cube roots of a number--- (1)

Thus resolvent has six (3! = 6) roots. Plugging in these values on obtain only three roots of a cubic equation since each of the cubic roots is obtained from two of the resolvent roots. These solutions are

--- (1)

Where r1 and r2 are the solutions of the resolvent which in this case is six degree polynomial. So Lagrange found a general theorem for the resolvent and its connection with the roots of the original polynomial. Each root of the resolvent polynomial can be expressed as the linear combination of the roots of the original polynomial

--- (1)

## RANDOM WALK

Random walk is a mathematical object that describes a path that are made up with a succession of random steps taken on a space like the set of integers. It can be explained using random variables. As it is a random process it follows bel curve of Gauss (gaussian distribution). Let us explore more of random walk by an example of coin flip. If we flip a coin randomly it will have two outcomes or being either head(H) or tail(T). If it is head (H) the value of random step is 1 and if it is tail the value of radom step is -1. So after two trial if its H and T the value of radom walk will be -1 + 1 = 0. Using this logic we can generate a random walk of infinite size. A finite size random walk will beThis was an example of random walk in one-dimesional space. We can create similar walk on higher dimensional space using the same argument as before.