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Remainder Theorem

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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)

Algebraic Element

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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 overK, 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 Kthe 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 s…

Root of Polynomials

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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
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 an…

Rings With Unity

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Field Adjunctions

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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 …

Subfields and Field Extensions

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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

                                       …

Field

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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 e…

Permutations

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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…