## Tuesday, 26 November 2013

### Remainder Theorem

If a polynomials   f(x)   is divided by    (x-a)   i.e. a polynomial of degree  1   then the remainder is    f(a)   .

We know that
f(x) = g(x) q(x) + r(x)

where degree    r(x) <  degree   g(x)

choose     g(x) = (x-a)

there fore    f(x) =  (x-a)  q(x)  +  r(x)

where degree  r(x) <  degree g(x)  ,   i.e.  <1  ,  or   degree  r(x) =0   or   say   r(x) = r  .

therefore ,   f(x) = (x-a) q(x) + r

therefore ,   f(a) = (a-a) q(a) + r

or   f(a) = r =  remainder when the polynomial  f(x)   is divided by   x-a     .

therefore  ,     f(x) = (x-a) q(x) +f(a)

Here is an example of Reminder Theorem

Show that the polynomial remainder theorem holds for an arbitrary second degree polynomial $f(x) = ax^2 + bx + c$ by using algebraic manipulation:
\begin{align} \frac{f(x)}{{x - r}} &= \frac{{a{x^2} + bx + c}}{{x - r}} \\ &= \frac{{ax(x - r) + (b + ar)x + c}}{{x - r}} \\ &= ax + \frac{{(b + ar)(x - r) + c + r(b + ar)}}{{x - r}} \\ &= ax + b + ar + \frac{{c + r(b + ar)}}{{x - r}} \\ &= ax + b + ar + \frac{{a{r^2} + br + c}}{{x - r}} \end{align}
Multiplying both sides by (x − r) gives
$f(x) = ax^2 + bx + c = (ax + b + ar)(x - r) + {a{r^2} + br + c}$.
Since $R = ar^2 + br + c$ is our remainder, we have indeed shown that $f(r) = R$.

## Saturday, 23 November 2013

### Algebraic Element

Definition :-  Let   K   be an extension of a field    F ,     then an element   a   which is element of   K    is said to be Algebraic over   F  if there is a non - zero polynomial  p(x)  is element of  F[x]   for which    p(a) = 0 .

If L is a field extension of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial g(x) with coefficients in K such that g(a)=0. Elements of L which are not algebraic over K are called transcendental over K.
These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is C/Q, C being the field of complex numbers and Q being the field of rational numbers).

The following conditions are equivalent for an element a of L:
• a is algebraic over K
• the field extension K(a)/K has finite degree, i.e. the dimension of K(a) as a K-vector space is finite. (Here K(a) denotes the smallest subfield of L containing K and a)
• K[a] = K(a), where K[a] is the set of all elements of L that can be written in the form g(a) with a polynomial g whose coefficients lie in K.
This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over K are again algebraic over K. The set of all elements of L which are algebraic over K is a field that sits in between L and K.
If a is algebraic over K, then there are many non-zero polynomials g(x) with coefficients in K such that g(a) = 0. However there is a single one with smallest degree and with leading coefficient 1. This is the minimal polynomial of a and it encodes many important properties of a.
Fields that do not allow any algebraic elements over them (except their own elements) are called algebraically closed. The field of complex numbers is an example.

Transcendental Element
Definition :-  Let   K    be an extension over  F   then an element  a   which is an element of    K    is said to be Transcendental over   F    if it is not Algebraic over   F    .

Hence we can say that  A complex number is said to be an Algebraic Number if it is Algebraic over the field of Rational Numbers . In other words it means that it satisfies as Polynomial Equation with rational coefficients not all zero .
A complex number which is not algebraic is called transcendental  e.g.    Pi    and   e     are transcendental numbers .

## Wednesday, 20 November 2013

### Root of Polynomials

Definition :-   Let  F  be any field and  p(x)  be any polynomial in   F[x]   . Then an element    a    lying in some extension field of    F   is called a root of   p(x)   if   p(a) = 0   .

from the theorems of polynomials we know that the element lying in an extension    K    of    F    which were algebraic over   F   i.e. which satisfies a polynomial

p(x)  is element of   F[x]    i.e.    p(a) = 0

Here in this article we shall aim at finding an extension field    K   of   F    in which a given polynomial

p(x)   is an element of   F[x]   has a root .

The statistical properties of the roots of a random polynomial have been the subject of several studies. Let
$p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$
be a random polynomial. If the coefficients ai are independently and identically distributed with a mean of zero, the real roots are mostly located near ±1. The complex roots can be shown to be on or close to the unit circle.
If the coefficients are Gaussian distributed with a mean of zero and variance of σ then the mean density of real roots is given by the Kac formula
$m( x ) = \frac { \sqrt{ A( x ) C( x ) - B( x )^2 }} {\pi A( x )}$
where
\begin{align} A( x ) &= \sigma \sum { x^{ 2i } } = \sigma \frac{ x^{ 2n } - 1 } { x - 1 }, \\ B( x ) &= \frac{ 1 } { 2 } \frac{ d } { dt } A( x ), \\ C( x ) &= \frac{ 1 } { 4 } \frac{ d^2 } { dt^2 } A( x ) + \frac{ 1 } { 4x } \frac{ d } { dt } A( x ). \end{align}
When the coefficients are Gaussian distributed with a non zero mean and variance of σ, a similar but more complex formula is known .

## Monday, 18 November 2013

### Rings With Unity

The Ring ( R,  +,  .  ) is said to be a Ring with Unity if it contains an element denoted by 1  such that
a  .  1  =  1  .  a
if and only if     a    is element of    .

The unit element  1  is called multiplicative identity . It should not be confused with integer  1  though both are denoted by the both symbol .

In the ring of integers the unit element is the integer  1  whereas in the ring of matrices the unit element is the unit matrix of suitable order .  In the ring of all even integers there is no unit element and as such it is a Ring without Unity .

Similarly the function   e   denoted by    e(x) =  1   if and only if    x   is the element of    [ 0 , 1 ]  is the unit element because in this case  .

( ef ) x =  e(x)  f(x) = 1 . f(x) = f(x)
then fore   ef = f
Similarly , (fe) x = f(x) e(x) = f(x) .1  = f(x)
then fore   fe = f

We may also understand it by this discussion

Formally, a ring is an Abelian group (R, +), together with a second binary operation * such that for all a, b and c in R,
$a * (b*c) = (a*b) * c$
$a * (b+c) = (a*b) + (a*c)$
$(a+b) * c = (a*c) + (b*c)$
also, if there exists a multiplicative identity in the ring, that is, an element e such that for all a in R,
$a*e = e*a = a$
then it is said to be a ring with unity or a unitary ring. The unity is often denoted 1, since the number 1 is the unity in the common rings of numbers.
The ring in which e is equal to the additive identity must have only one element. This ring is called the trivial ring.
Rings that lie within other rings are called subrings. Maps between rings which respect the ring operations are called ring homomorphisms. Rings, together with ring homomorphisms, form a category (the category of rings). Closely related is the notion of ideals, certain subsets of rings which arise as kernels of homomorphisms and can serve to define factor rings. Basic facts about ideals, homomorphisms and factor rings are recorded in the isomorphism theorems and in the Chinese remainder theorem.

Noncommutative rings resemble rings of matrices in many respects. Following the model of algebraic geometry, attempts have been made recently at defining noncommutative geometry based on noncommutative rings. Noncommutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules. A module over a ring is an Abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces. Examples of noncommutative rings are given by rings of square matrices or more generally by rings of endomorphisms of Abelian groups or modules, and by monoid rings.

If anyone has any questions or wants to discuss more please write your comments and start a good discussion .

## Monday, 11 November 2013

Let   F  b a given field  K  be an extension of   F  and   a   is the  element of   K  .  Suppose   c  is the collection o all sub field of  K  which contain both  a  and  F  . Evidently   c  is non empty as at least   K  itself   ( containing both   a  and   F )  belongs to it .

Denoted by   F(a)  the intersection o all these sub fields of  K  which are members of   c  then   F(a)   is also a sub fields because we know that the intersection of an arbitrary collection of sub fields of   K  is also a sub field of    K    .

The sub field   F(a)   contains both   F  and    a   as every member of  c  contains both  F  and  a  and hence by definition   F(a)    is an element of   c   .

Again   F(a)  being the intersection of   all members of   c   , it therefore must be contained in every member of    c   .

Hence we conclude that   F(a)  is a sub field of  K   containing both   F   and   a   and itself contained in any sub field of   K   containing  both    F   and   a   (i.e. contain every member of    c  ) .  Therefore   F(a)  is the smallest  sub field of     K    containing both   F   and  a   , an is obtained by adjoining an element   a  of field   K   of its sub field    F   . The above process of adjoining an element of a field to its sub field is known a Field Adjunction  .

Let E be a field extension of a field F. Given a set of elements A in the larger field E we denote by F(A) the smallest subextension which contains the elements of A. We say F(A) is constructed by adjunction of the elements A to F or generated by A.
If A is finite we say F(A) is finitely generated and if A consists of a single element we say F(A) is a simple extension. The primitive element theorem states a finite separable extension is simple.
In a sense, a finitely generated extension is a transcendental generalization of a finite extension since, if the generators in A are all algebraic, then F(A) is a finite extension of F. Because of this, most examples come from algebraic geometry.
A subextension of a finitely generated field extension is also a finitely generated extension.

Given a field extension E/F and a subset A of E, let $\mathcal{T}$ be the family of all finite subsets of A. Then
$F(A) = \bigcup_{T \in \mathcal{T}} F(T)$.
In other words the adjunction of any set can be reduced to a union of adjunctions of finite sets.
Given a field extension E/F and two subsets N, M of E then K(MN) = (K(M))(N) = (K(N))(M). This shows that any adjunction of a finite set can be reduced to a successive adjunction of single elements.

F(A) consists of all those elements of E that can be constructed using a finite number of field operations +, -, *, / applied to elements from F and A. For this reason F(A) is sometimes called the field of rational expressions in F and A.

## Friday, 8 November 2013

### Subfields and Field Extensions

Before start to know about Subfields  we have to think some thing more about Field .

In case   F  be a field then the set of all Polynomials   F[x]   is an an integral domain .

Choose   f(x) ,  g(x)  to any two elements of   F[x]  ,  then   f(x)  is said to be a divisor of   g(x)  if there exists a polynomial   h(x)   in   F[x]  such that .

g(x)  =  f(x) h(x)  .

The statement   f(x)   is a divisor of   g(x)  is expressed symbolically as   f(x)/g(x)   .

In case   f(x)/g(x)  and   g(x)/f(x)   ,  then both   f(x)  and  g(x)  will be termed as associates and in the case   f(x) = c.g(x)   for some  c is not equivalent to zero that is element of  F  .

Unit of F[x]  :-    A unit of   F[x]   is that element which have multiplicative inverse and we have shown above that in the polynomials   F[x]   the only   (multiplicatively) elements are constant polynomials .

Hence all constant polynomials in

F[x]  are units of   F[x]  .

### Subfields and field extensions

A subfield is, informally, a small field contained in a bigger one. Formally, a subfield E of a field F is a subset containing 0 and 1, closed under the operations +, −, · and multiplicative inverses and with its own operations defined by restriction. For example, the real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers and the rational numbers are examples.
The notion of field extension lies at the heart of field theory, and is crucial to many other algebraic domains. A field extension F / E is simply a field F and a subfield EF. Constructing such a field extension F / E can be done by "adding new elements" or adjoining elements to the field E. For example, given a field E, the set F = E(X) of rational functions, i.e., equivalence classes of expressions of the kind
$\frac{p(X)}{q(X)},$
where p(X) and q(X) are polynomials with coefficients in E, and q is not the zero polynomial, forms a field. This is the simplest example of a transcendental extension of E. It also is an example of a domain (the ring of polynomials $\scriptstyle E$ in this case) being embedded into its field of fractions $\scriptstyle E(X)$.
The ring of formal power series $\scriptstyle E[[X]]$ is also a domain, and again the (equivalence classes of) fractions of the form p(X)/ q(X) where p and q are elements of $\scriptstyle E[[X]]$ form the field of fractions for $\scriptstyle E[[X]]$. This field is actually the ring of Laurent series over the field E, denoted $\scriptstyle E((X))$.
In the above two cases, the added symbol X and its powers did not interact with elements of E. It is possible however that the adjoined symbol may interact with E. This idea will be illustrated by adjoining an element to the field of real numbers R. As explained above, C is an extension of R. C can be obtained from R by adjoining the imaginary symbol i which satisfies i2 = −1. The result is that R[i]=C. This is different from adjoining the symbol X to R, because in that case, the powers of X are all distinct objects, but here, i2=−1 is actually an element of R.
Another way to view this last example is to note that i is a zero of the polynomial p(X) = X2 + 1. The quotient ring $\scriptstyle R[X]/(X^2 \,+\, 1)$ can be mapped onto C using the map $\scriptstyle \overline{a \,+\, bX} \;\rightarrow\; a \,+\, ib$. Since the ideal (X2+1) is generated by a polynomial irreducible over R, the ideal is maximal, hence the quotient ring is a field. This nonzero ring map from the quotient to C is necessarily an isomorphism of rings.
The above construction generalises to any irreducible polynomial in the polynomial ring E[X], i.e., a polynomial p(X) that cannot be written as a product of non-constant polynomials. The quotient ring F = E[X] / (p(X)), is again a field.

There are many topics of Field and Subfield for discussion . If anyone wants to discuss more about Field and Subfield  may post their comments for a good discussion .

## Wednesday, 6 November 2013

### Field

Definition :-  A Ring ( R , + , . ) which has at least two elements is called a field  if
a)     It is a commutative ring
b)     It is a ring with unity
c)     All non zero elements are inversible  w . r. t . multiplication .

i.e.    a   is not equivalent to zero then  b   in   R  such that  ab = ba = 1  ( unity of the ring ) then b= 1/a i.e. multiplicative inverse of   a  .

The integral domain and field are both commutative rings with unity and their third property is different , i.e. for I. D. it is a ring without zero divisors and for a field it is a ring in which all non zero elements are inversible.

Alternative Definition :-    Combining the above properties we can give an alternate definition of a field as below .

A Ring ( R , + , . ) with at least two elements is called a Field if its non zero elements from an abelian group under multiplication .

The condition  R2  for a ring proves closure and associativity for multiplication , The ring is with unity shows the existence of multiplicative  identity . The commutative property proves the character of an abelian group .
All non zero elements having their inverses prove the existence of inverses . Hence the above alternative definition .

Intuitively, a field is a set F that is a commutative group with respect to two compatible operations, addition and multiplication, with "compatible" being formalized by distributivity, and the caveat that the additive identity (0) has no multiplicative inverse (one cannot divide by 0).
The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold; subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication:
Closure of F under addition and multiplication
For all a, b in F, both a + b and a · b are in F (or more formally, + and · are binary operations on F).
For all a, b, and c in F, the following equalities hold: a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c.
For all a and b in F, the following equalities hold: a + b = b + a and a · b = b · a.
Existence of additive and multiplicative identity elements
There exists an element of F, called the additive identity element and denoted by 0, such that for all a in F, a + 0 = a. Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all a in F, a · 1 = a. To exclude the trivial ring, the additive identity and the multiplicative identity are required to be distinct.
Existence of additive inverses and multiplicative inverses
For every a in F, there exists an element −a in F, such that a + (−a) = 0. Similarly, for any a in F other than 0, there exists an element a−1 in F, such that a · a−1 = 1. (The elements a + (−b) and a · b−1 are also denoted a − b and a/b, respectively.) In other words, subtraction and division operations exist.
For all a, b and c in F, the following equality holds: a · (b + c) = (a · b) + (a · c).
A field is therefore an algebraic structure 〈F, +, ·, −, −1, 0, 1〉; of type 〈2, 2, 1, 1, 0, 0〉, consisting of two abelian groups:
• F under +, −, and 0;
• F \ {0} under ·, −1, and 1, with 0 ≠ 1,
with · distributing over +.

## Saturday, 2 November 2013

### Permutations

Consider the element   a  ,  b  ,  c   of a set   {  a ,  b ,  c  }  .
We can arrange these letters   a ,  b ,  c ,  in the following six manners .

a , b, c ;  a , c , b ;  b , c , a  ;   c , a , b  ;  c , b , a  ;

i.e. there are 3  !  =  6   ways of arranging them or there are   3  !    permutations of the three elements   a ,  b ,  c  .   The permutations are written as  P1   P2  ............. ,  and we adopt a two line notations to express the permutations .

In the first line we write the element in their natural order and in the line below it we write them in the order in which they have been arranged .

i. e.         P1  =  (  a  b  c  )
(  a  b  c  )

i.e.   a    ---->   a     ,   b  ---->  b   ,     c  ----->  c     ,

There is no change and this type of permutation is called   Identity  Permutation  and is written as  l   .

We can also understand Permutations as follows  ---::

Permutation is used with several slightly different meanings, all related to the act of permuting (rearranging) objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). For example, an anagram of a word is a permutation of its letters. The study of permutations in this sense generally belongs to the field of combinatorics.
The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×⋯×1, which is commonly denoted as "n factorial" and written "n!".
Permutations occur, in more or less prominent ways, in almost every domain of mathematics. They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons permutations arise in the study of sorting algorithms in computer science.
In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself (i.e., a map SS for which every element of S occurs exactly once as image value). This is related to the rearrangement of S in which each element s takes the place of the corresponding f(s). The collection of such permutations form a symmetric group. The key to its structure is the possibility to compose permutations: performing two given rearrangements in succession defines a third rearrangement, the composition. Permutations may act on composite objects by rearranging their components, or by certain replacements (substitutions) of symbols.
In elementary combinatorics, the k-permutations, or partial permutations, are the sequences of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.

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