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Hello friends welcome back in this tutorial we will study about algebraic structures so the combination of the set and the operations that are applied on the elements of the set is called an algebraic structure here we will study three common algebraic structures groups rings and fields so first of all we will
Study about groups a group Chi is a set of elements with a binary operation star that satisfies these four properties first one is closure property but so according to this closure property what happens if a and B are elements of Group C then whatever the results see is
Produced after applying this a binary operation star on a and B then that result should also be an element of columns G second is associative any property according to this property if a B and C are elements of group G then this equation must be true now the third
Property is existence of an identity according to this property for all elements a of group G there exists an element E which is called the identity element and when we perform the binary operation star between this identity element E and any element a of curve G
The result will always be the element E so according to this property existence of identity what happens so for all the elements a which are in Group G for all these elements a there exists an identity element E so when we perform a binary operation star between this
Identity element E and any element a of Grouper G we will get a as result fourth property’s existence of an inverse according to this property for each element a in Group G there exists an element a – which is called the inverse of element E and when we combine the
Operation is start between a and a a – the result will be an identity element e right so according to this property what happens there for each element a of fukuchi there is an inverse and when we perform the binary operation star between the element and its inverse the result will
Be an identity element now next topic is abelian group so what happens if a group also satisfies one extra property called commutativity then that group is called opinion group right so what is this a commutativity property according to this property if we a and B are elements in G
Then a star B should be equal to B star II right so according to this property community activity for all elements a and B in G we have a star B is equal to B star D so if any group which satisfies this comment relativity property also
Then that group is called a billion group and abelian group is also called from it a difficult now let us study about rink so a range denoted as R is equal to this here this represents is set and this is star and rectangle they represent binary operations so even
Has a set and to find reoperation so ring is a an algebraic structure with dessert and 2.0 operations which are represented here by a star and rectangle so the first operation of ring must satisfy all five properties of the opinion that is closure property associative ‘ti commutativity existence
Of identity and existence of inverse so first operation of ring must satisfy these five properties now the second operation of the ring must satisfy only these two properties closure property and associativity and another thing is that the second operation which is represented here by a rectangle it must
Be distributive over the first operation the star so what is the meaning of that this the second operation rectangle is distributive over the first operation star it means that for all elements a B and C of are these two equations must hold so when these two equations are
Whole and then it means that a second operation rectangle is distributive over the first operation start so a rink in which the commutative property is also satisfied for the second operation is called a commutative ring now we will study about filled so if filled is denoted as like this so
Here this is a set and star and rectangle they represent two binary operations so if will def is a commutative ring in which the second operation satisfies all five properties defined for the first operation except that the identity element of the first operation has no inverse with respect to
The second operation a finite field is a filled with finite number of elements gallery showed that for a filled to be finite the number of elements should be P raised to power n where P is a prime and N is a positive integer the finite fields are usually called alloys filled
And denoted as G of P raised to power n AG alloys for G of P raised to power n is a finite filled with purest power n elements now in this folding you can see G of P raised to power n when and is
Equal to 1 then we have g FP filled now consider this that is ID p this such that p as i told you in previous tutorial also it has elements 0 1 2 3 so on up to P minus 1 so this such that p is a gfp filled with two operations
Addition modulo P and multiplication modulo P now let us see an example now let us see this example said seven here seven is a prime number so that someone is a GFP filled with two operations addition modulo 7 and multiplication modulus F so here we are defining G of 7
On chat 7 with two operations like so here in this table you can see that this table shows the result of addition modulo 7 operation on every pair of elements of set set 7 and here in this table you can see the result of modulo 7 operation on every pair of elements of
Sector 7 so when you observe this table which shows the result of addition modulo 7 operation a big pair of ailments offsets at 7:00 then you can see that the identity element is zero because when you affirm addition modulo 7 operation between 0 and any other element of such that 7 the
Result will be the other element offsets at 7 so 0 is the identity element for this operation addition modulo 7 now see in this table in this table you can see that the identity element is 1 in this table you can see that the identity element is 1 for this operation
Multiplication modulo 7 because when you perform multiplication modulo 7 operation between 1 and any other element of set 7 then what happens you will get the other element offset judge 7 as an result right you will get the other element of set aside 7 as a phizzer therefore what happens to the
Identity element identity element with respect to this operation multiplication modulo 7 is 1 now see this tables now this table what this table does this table shows the inverse of every element of satis at 7 with respect to the addition modulo 7 operation and the multiplication models have an operation
So when you see this table you can see that every element of set 7 as an inverse with respect to addition modulo 7 operation but only the identity element of addition modulo 7 operation right only this identity element that is 0 right identity element of addition modulo 7 operation it does not have
Inverse with respect to multiplication moly lose 7 operation as you can see here in this table right but other elements of set 7 they have the inverse with respect to the multiplication modulo 7 operation right
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