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Page 1
Edurev123
3. Fields
3.1 Consider the polynomial ring ?? [?? ]. Show ?? (?? )=?? ?? -?? is irreducible over ?? .
Let ?? be the ideal in ?? [?? ] generated by ?? (?? ) . Then show that ?? [?? ]/?? is a field and
that each element is of the form ?? ?? +?? ?? ?? +?? ?? ?? ?? with ?? ,?? ?? ,?? ?? in ?? and ?? =?? +?? .
(2010 : 15 Marks)
Solution:
Given :
?? (?? )=?? 3
-2
Let ?? (?? ),?? (?? )??? [?? ], and ?? (?? ) is reducible such that
?? (?? ) =?? (?? )·?? (?? )
Suppose, ?? (?? ) =?? +?? ? ?? (?? ) =?? 2
+?? ?? +?? ???? (?? ) =?? 3
-2=(?? +?? )(?? 2
+???? +?? )
? ?? 3
-2 =?? 3
+?? ?? 2
+???? +?? ?? 2
+???? +????
Comparing LHS & RHS, we get
?? +?? ?=0??? =-?? ???? +?? ?=0
??-?? 2
+?? ?=0??? =?? 2
???? ?=-2
???? 3
?=-2??? =-2
1/3
which is not a rational number
i.e., ?? ??? ??? ,?? ???
??? (?? ) is irreducible over ?? .
Now, ?? is ideal generated by ?? (?? )
i.e.,
?? =??? (?? )?
?
?? (?? )
?? is a field as ?? (?? ) is irreducible (theorem)
Page 2
Edurev123
3. Fields
3.1 Consider the polynomial ring ?? [?? ]. Show ?? (?? )=?? ?? -?? is irreducible over ?? .
Let ?? be the ideal in ?? [?? ] generated by ?? (?? ) . Then show that ?? [?? ]/?? is a field and
that each element is of the form ?? ?? +?? ?? ?? +?? ?? ?? ?? with ?? ,?? ?? ,?? ?? in ?? and ?? =?? +?? .
(2010 : 15 Marks)
Solution:
Given :
?? (?? )=?? 3
-2
Let ?? (?? ),?? (?? )??? [?? ], and ?? (?? ) is reducible such that
?? (?? ) =?? (?? )·?? (?? )
Suppose, ?? (?? ) =?? +?? ? ?? (?? ) =?? 2
+?? ?? +?? ???? (?? ) =?? 3
-2=(?? +?? )(?? 2
+???? +?? )
? ?? 3
-2 =?? 3
+?? ?? 2
+???? +?? ?? 2
+???? +????
Comparing LHS & RHS, we get
?? +?? ?=0??? =-?? ???? +?? ?=0
??-?? 2
+?? ?=0??? =?? 2
???? ?=-2
???? 3
?=-2??? =-2
1/3
which is not a rational number
i.e., ?? ??? ??? ,?? ???
??? (?? ) is irreducible over ?? .
Now, ?? is ideal generated by ?? (?? )
i.e.,
?? =??? (?? )?
?
?? (?? )
?? is a field as ?? (?? ) is irreducible (theorem)
???
?? (?? )
?? =??? (?? )>+?? 0
+?? 1
?? +?? 2
?? 2
=?? +?? 0
+?? 1
?? +?? 2
?? 2
deg·(?? 0
+?? 1
?? +?? 2
?? 2
)<deg·(?? (?? ))
?????????????????????
remainder term
where ?? 0
,?? 1
,?? 2
??? . Now,
?? ?=?? +?? ?? 0
+?? 1
?? +?? 2
?? 2
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? +?? )
2
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? +?? )(?? +?? )
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? 2
+?? )
?=?? 0
+?? 1
?? +?? 2
?? 2
+?? which is equivalent to eqn. (1)
?
?? (?? )
?? can be written as ?? 0
+?? 1
?? +?? 2
2
2
where ?? =?? +?? .
3.2 Show that the set of matrices ?? ={[
?? -?? ?? ?? ]?? ,?? ??? } is a field under the usual
binary operations of matrix addition and matrix multiplication. What are the
additive and multiplicative identities and what is the inverse of [
?? -?? ?? ?? ] ? Consider
iiie map ?? :?? ??? defined by ?? (?? +???? )=[
?? -?? ?? ?? ]. Show that ?? is an isomorphism
(Here ?? is the set of real numbers and ?? is the set of complex numbers)?
(2013 : 10 Marks)
Solution:
Approach : Prove only the important parts in actual exam.
SCM?(2,?? ) where ?? (?? ,?? ) is the ring of 2×2 matrices with entries from ?? .
To show ?? is a field
and
[
?? -?? ?? ?? ]??? ?[
?? -?? ?? ?? ]
-1
=[
?? ?? 2
+?? 2
?? ?? 2
+?? 2
-?? ?? 2
+?? 2
?? ?? 2
+?? 2
]
-1
??? [
?? -?? ?? ?? ][
?? -?? ?? ?? ]
-1
=[
?? -?? ?? ?? ][
?? ?? 2
+?? 2
?? ?? 2
+?? 2
-?? ?? 2
+?? 2
?? ?? 2
+?? 2
]=[
???? +????
?? 2
+?? 2
-(???? -???? )
?? 2
+?? 2
???? -????
?? 2
+?? 2
???? +????
?? 2
+?? 2
]???
So, ?? is closed w.r.t. multiplication and has multiplicative inverse.
Also,
Page 3
Edurev123
3. Fields
3.1 Consider the polynomial ring ?? [?? ]. Show ?? (?? )=?? ?? -?? is irreducible over ?? .
Let ?? be the ideal in ?? [?? ] generated by ?? (?? ) . Then show that ?? [?? ]/?? is a field and
that each element is of the form ?? ?? +?? ?? ?? +?? ?? ?? ?? with ?? ,?? ?? ,?? ?? in ?? and ?? =?? +?? .
(2010 : 15 Marks)
Solution:
Given :
?? (?? )=?? 3
-2
Let ?? (?? ),?? (?? )??? [?? ], and ?? (?? ) is reducible such that
?? (?? ) =?? (?? )·?? (?? )
Suppose, ?? (?? ) =?? +?? ? ?? (?? ) =?? 2
+?? ?? +?? ???? (?? ) =?? 3
-2=(?? +?? )(?? 2
+???? +?? )
? ?? 3
-2 =?? 3
+?? ?? 2
+???? +?? ?? 2
+???? +????
Comparing LHS & RHS, we get
?? +?? ?=0??? =-?? ???? +?? ?=0
??-?? 2
+?? ?=0??? =?? 2
???? ?=-2
???? 3
?=-2??? =-2
1/3
which is not a rational number
i.e., ?? ??? ??? ,?? ???
??? (?? ) is irreducible over ?? .
Now, ?? is ideal generated by ?? (?? )
i.e.,
?? =??? (?? )?
?
?? (?? )
?? is a field as ?? (?? ) is irreducible (theorem)
???
?? (?? )
?? =??? (?? )>+?? 0
+?? 1
?? +?? 2
?? 2
=?? +?? 0
+?? 1
?? +?? 2
?? 2
deg·(?? 0
+?? 1
?? +?? 2
?? 2
)<deg·(?? (?? ))
?????????????????????
remainder term
where ?? 0
,?? 1
,?? 2
??? . Now,
?? ?=?? +?? ?? 0
+?? 1
?? +?? 2
?? 2
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? +?? )
2
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? +?? )(?? +?? )
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? 2
+?? )
?=?? 0
+?? 1
?? +?? 2
?? 2
+?? which is equivalent to eqn. (1)
?
?? (?? )
?? can be written as ?? 0
+?? 1
?? +?? 2
2
2
where ?? =?? +?? .
3.2 Show that the set of matrices ?? ={[
?? -?? ?? ?? ]?? ,?? ??? } is a field under the usual
binary operations of matrix addition and matrix multiplication. What are the
additive and multiplicative identities and what is the inverse of [
?? -?? ?? ?? ] ? Consider
iiie map ?? :?? ??? defined by ?? (?? +???? )=[
?? -?? ?? ?? ]. Show that ?? is an isomorphism
(Here ?? is the set of real numbers and ?? is the set of complex numbers)?
(2013 : 10 Marks)
Solution:
Approach : Prove only the important parts in actual exam.
SCM?(2,?? ) where ?? (?? ,?? ) is the ring of 2×2 matrices with entries from ?? .
To show ?? is a field
and
[
?? -?? ?? ?? ]??? ?[
?? -?? ?? ?? ]
-1
=[
?? ?? 2
+?? 2
?? ?? 2
+?? 2
-?? ?? 2
+?? 2
?? ?? 2
+?? 2
]
-1
??? [
?? -?? ?? ?? ][
?? -?? ?? ?? ]
-1
=[
?? -?? ?? ?? ][
?? ?? 2
+?? 2
?? ?? 2
+?? 2
-?? ?? 2
+?? 2
?? ?? 2
+?? 2
]=[
???? +????
?? 2
+?? 2
-(???? -???? )
?? 2
+?? 2
???? -????
?? 2
+?? 2
???? +????
?? 2
+?? 2
]???
So, ?? is closed w.r.t. multiplication and has multiplicative inverse.
Also,
[
?? -?? ?? ?? ][
?? -?? ?? ?? ]=[
???? -???? -(???? +???? )
???? +???? ???? -????
]=[
?? -?? ?? ?? ][
?? -?? ?? ?? ]
i.e., multiplication is commutative.
That addition is commutative, closed and has inverse follows from ?? being subset of
?? (2,?? ) . ??? is a field.
Additive identity :
[
?? -?? ?? ?? ]+[
0 0
0 0
]=[
?? -?? ?? ?? ]=[
0 0
0 0
]+[
?? -?? ?? ?? ]
?[
0 0
0 0
]??? is additive identity.
Similarly, [
1 0
0 1
]=?? 2
is multiplicative identity.
[
1 -1
1 1
]
-1
=[
1
2
1
2
-1
2
1
2
]??? ?? (?? +???? )=[
?? -?? ?? ?? ]
To show ?? is isomorphism, ?? is linear.
?? (?? +???? )+(?? +???? )]?=?? (?? +?? +??(?? +?? )]
?=[
?? +?? -(?? +?? )
?? +?? ?? +?? ]=[
?? -?? ?? ?? ]+[
?? -?? ?? ?? ]
3.3 Show that ?? ?? is a field. Then find ([?? ]+[?? ])
-?? and (-[?? ])
-?? in ?? ?? .
(2014 : 15 Marks)
Solution:
?? 7
={[0],[1],[2],[3],[4],[5],[6]}
For ?? 7
to be a field, it should satisfy
(i) (?? 7
,+) is an abelian.
(ii) (?? 7
,?? ) is an abelian group, where ?? 7
*
=?? 7
-{0}
(iii) Distributive law.
(i) (?? 7
+)
+ [0] [1] [2] [3] [4] [5] [6]
Page 4
Edurev123
3. Fields
3.1 Consider the polynomial ring ?? [?? ]. Show ?? (?? )=?? ?? -?? is irreducible over ?? .
Let ?? be the ideal in ?? [?? ] generated by ?? (?? ) . Then show that ?? [?? ]/?? is a field and
that each element is of the form ?? ?? +?? ?? ?? +?? ?? ?? ?? with ?? ,?? ?? ,?? ?? in ?? and ?? =?? +?? .
(2010 : 15 Marks)
Solution:
Given :
?? (?? )=?? 3
-2
Let ?? (?? ),?? (?? )??? [?? ], and ?? (?? ) is reducible such that
?? (?? ) =?? (?? )·?? (?? )
Suppose, ?? (?? ) =?? +?? ? ?? (?? ) =?? 2
+?? ?? +?? ???? (?? ) =?? 3
-2=(?? +?? )(?? 2
+???? +?? )
? ?? 3
-2 =?? 3
+?? ?? 2
+???? +?? ?? 2
+???? +????
Comparing LHS & RHS, we get
?? +?? ?=0??? =-?? ???? +?? ?=0
??-?? 2
+?? ?=0??? =?? 2
???? ?=-2
???? 3
?=-2??? =-2
1/3
which is not a rational number
i.e., ?? ??? ??? ,?? ???
??? (?? ) is irreducible over ?? .
Now, ?? is ideal generated by ?? (?? )
i.e.,
?? =??? (?? )?
?
?? (?? )
?? is a field as ?? (?? ) is irreducible (theorem)
???
?? (?? )
?? =??? (?? )>+?? 0
+?? 1
?? +?? 2
?? 2
=?? +?? 0
+?? 1
?? +?? 2
?? 2
deg·(?? 0
+?? 1
?? +?? 2
?? 2
)<deg·(?? (?? ))
?????????????????????
remainder term
where ?? 0
,?? 1
,?? 2
??? . Now,
?? ?=?? +?? ?? 0
+?? 1
?? +?? 2
?? 2
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? +?? )
2
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? +?? )(?? +?? )
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? 2
+?? )
?=?? 0
+?? 1
?? +?? 2
?? 2
+?? which is equivalent to eqn. (1)
?
?? (?? )
?? can be written as ?? 0
+?? 1
?? +?? 2
2
2
where ?? =?? +?? .
3.2 Show that the set of matrices ?? ={[
?? -?? ?? ?? ]?? ,?? ??? } is a field under the usual
binary operations of matrix addition and matrix multiplication. What are the
additive and multiplicative identities and what is the inverse of [
?? -?? ?? ?? ] ? Consider
iiie map ?? :?? ??? defined by ?? (?? +???? )=[
?? -?? ?? ?? ]. Show that ?? is an isomorphism
(Here ?? is the set of real numbers and ?? is the set of complex numbers)?
(2013 : 10 Marks)
Solution:
Approach : Prove only the important parts in actual exam.
SCM?(2,?? ) where ?? (?? ,?? ) is the ring of 2×2 matrices with entries from ?? .
To show ?? is a field
and
[
?? -?? ?? ?? ]??? ?[
?? -?? ?? ?? ]
-1
=[
?? ?? 2
+?? 2
?? ?? 2
+?? 2
-?? ?? 2
+?? 2
?? ?? 2
+?? 2
]
-1
??? [
?? -?? ?? ?? ][
?? -?? ?? ?? ]
-1
=[
?? -?? ?? ?? ][
?? ?? 2
+?? 2
?? ?? 2
+?? 2
-?? ?? 2
+?? 2
?? ?? 2
+?? 2
]=[
???? +????
?? 2
+?? 2
-(???? -???? )
?? 2
+?? 2
???? -????
?? 2
+?? 2
???? +????
?? 2
+?? 2
]???
So, ?? is closed w.r.t. multiplication and has multiplicative inverse.
Also,
[
?? -?? ?? ?? ][
?? -?? ?? ?? ]=[
???? -???? -(???? +???? )
???? +???? ???? -????
]=[
?? -?? ?? ?? ][
?? -?? ?? ?? ]
i.e., multiplication is commutative.
That addition is commutative, closed and has inverse follows from ?? being subset of
?? (2,?? ) . ??? is a field.
Additive identity :
[
?? -?? ?? ?? ]+[
0 0
0 0
]=[
?? -?? ?? ?? ]=[
0 0
0 0
]+[
?? -?? ?? ?? ]
?[
0 0
0 0
]??? is additive identity.
Similarly, [
1 0
0 1
]=?? 2
is multiplicative identity.
[
1 -1
1 1
]
-1
=[
1
2
1
2
-1
2
1
2
]??? ?? (?? +???? )=[
?? -?? ?? ?? ]
To show ?? is isomorphism, ?? is linear.
?? (?? +???? )+(?? +???? )]?=?? (?? +?? +??(?? +?? )]
?=[
?? +?? -(?? +?? )
?? +?? ?? +?? ]=[
?? -?? ?? ?? ]+[
?? -?? ?? ?? ]
3.3 Show that ?? ?? is a field. Then find ([?? ]+[?? ])
-?? and (-[?? ])
-?? in ?? ?? .
(2014 : 15 Marks)
Solution:
?? 7
={[0],[1],[2],[3],[4],[5],[6]}
For ?? 7
to be a field, it should satisfy
(i) (?? 7
,+) is an abelian.
(ii) (?? 7
,?? ) is an abelian group, where ?? 7
*
=?? 7
-{0}
(iii) Distributive law.
(i) (?? 7
+)
+ [0] [1] [2] [3] [4] [5] [6]
[0] [0] [1] [2] [3] [4] [5] [6]
[1] [1] [2] [3] [4] [5] [6] [0]
[2] [2] [3] [4] [5] [6] [0] [1]
[3] [3] [4] [5] [6] [0] [1] [2]
[4] [4] [5] [6] [0] [1] [2] [3]
[5] [5] [6] [0] [1] [2] [3] [4]
[6] [6] [0] [1] [2] [3] [4] [5]
? All elements belong to ?? 7
- closure property.
? First row coincide with the top row, then [0] is identity element.
? [0] is in every row & column
? Inverse property satisfied.
([?? ]+[?? ]+[?? ]=[?? +?? ]+[?? ]=[?? +?? +?? ]=[?? ]+[?? +?? ]=[?? ]+([?? ]+[?? ])
? Associative property satisfied.
? Transpose of matrix equal to original matrix.
? Commutative property satisfy.
(ii) (?? 7
*
,?? )
? Every element belong to ?? 7
*
-closure property.
? Top row coincide with first row, Hence [1] is an identity element.
Page 5
Edurev123
3. Fields
3.1 Consider the polynomial ring ?? [?? ]. Show ?? (?? )=?? ?? -?? is irreducible over ?? .
Let ?? be the ideal in ?? [?? ] generated by ?? (?? ) . Then show that ?? [?? ]/?? is a field and
that each element is of the form ?? ?? +?? ?? ?? +?? ?? ?? ?? with ?? ,?? ?? ,?? ?? in ?? and ?? =?? +?? .
(2010 : 15 Marks)
Solution:
Given :
?? (?? )=?? 3
-2
Let ?? (?? ),?? (?? )??? [?? ], and ?? (?? ) is reducible such that
?? (?? ) =?? (?? )·?? (?? )
Suppose, ?? (?? ) =?? +?? ? ?? (?? ) =?? 2
+?? ?? +?? ???? (?? ) =?? 3
-2=(?? +?? )(?? 2
+???? +?? )
? ?? 3
-2 =?? 3
+?? ?? 2
+???? +?? ?? 2
+???? +????
Comparing LHS & RHS, we get
?? +?? ?=0??? =-?? ???? +?? ?=0
??-?? 2
+?? ?=0??? =?? 2
???? ?=-2
???? 3
?=-2??? =-2
1/3
which is not a rational number
i.e., ?? ??? ??? ,?? ???
??? (?? ) is irreducible over ?? .
Now, ?? is ideal generated by ?? (?? )
i.e.,
?? =??? (?? )?
?
?? (?? )
?? is a field as ?? (?? ) is irreducible (theorem)
???
?? (?? )
?? =??? (?? )>+?? 0
+?? 1
?? +?? 2
?? 2
=?? +?? 0
+?? 1
?? +?? 2
?? 2
deg·(?? 0
+?? 1
?? +?? 2
?? 2
)<deg·(?? (?? ))
?????????????????????
remainder term
where ?? 0
,?? 1
,?? 2
??? . Now,
?? ?=?? +?? ?? 0
+?? 1
?? +?? 2
?? 2
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? +?? )
2
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? +?? )(?? +?? )
?=?? 0
+?? 1
(?? +?? )+?? 2
(?? 2
+?? )
?=?? 0
+?? 1
?? +?? 2
?? 2
+?? which is equivalent to eqn. (1)
?
?? (?? )
?? can be written as ?? 0
+?? 1
?? +?? 2
2
2
where ?? =?? +?? .
3.2 Show that the set of matrices ?? ={[
?? -?? ?? ?? ]?? ,?? ??? } is a field under the usual
binary operations of matrix addition and matrix multiplication. What are the
additive and multiplicative identities and what is the inverse of [
?? -?? ?? ?? ] ? Consider
iiie map ?? :?? ??? defined by ?? (?? +???? )=[
?? -?? ?? ?? ]. Show that ?? is an isomorphism
(Here ?? is the set of real numbers and ?? is the set of complex numbers)?
(2013 : 10 Marks)
Solution:
Approach : Prove only the important parts in actual exam.
SCM?(2,?? ) where ?? (?? ,?? ) is the ring of 2×2 matrices with entries from ?? .
To show ?? is a field
and
[
?? -?? ?? ?? ]??? ?[
?? -?? ?? ?? ]
-1
=[
?? ?? 2
+?? 2
?? ?? 2
+?? 2
-?? ?? 2
+?? 2
?? ?? 2
+?? 2
]
-1
??? [
?? -?? ?? ?? ][
?? -?? ?? ?? ]
-1
=[
?? -?? ?? ?? ][
?? ?? 2
+?? 2
?? ?? 2
+?? 2
-?? ?? 2
+?? 2
?? ?? 2
+?? 2
]=[
???? +????
?? 2
+?? 2
-(???? -???? )
?? 2
+?? 2
???? -????
?? 2
+?? 2
???? +????
?? 2
+?? 2
]???
So, ?? is closed w.r.t. multiplication and has multiplicative inverse.
Also,
[
?? -?? ?? ?? ][
?? -?? ?? ?? ]=[
???? -???? -(???? +???? )
???? +???? ???? -????
]=[
?? -?? ?? ?? ][
?? -?? ?? ?? ]
i.e., multiplication is commutative.
That addition is commutative, closed and has inverse follows from ?? being subset of
?? (2,?? ) . ??? is a field.
Additive identity :
[
?? -?? ?? ?? ]+[
0 0
0 0
]=[
?? -?? ?? ?? ]=[
0 0
0 0
]+[
?? -?? ?? ?? ]
?[
0 0
0 0
]??? is additive identity.
Similarly, [
1 0
0 1
]=?? 2
is multiplicative identity.
[
1 -1
1 1
]
-1
=[
1
2
1
2
-1
2
1
2
]??? ?? (?? +???? )=[
?? -?? ?? ?? ]
To show ?? is isomorphism, ?? is linear.
?? (?? +???? )+(?? +???? )]?=?? (?? +?? +??(?? +?? )]
?=[
?? +?? -(?? +?? )
?? +?? ?? +?? ]=[
?? -?? ?? ?? ]+[
?? -?? ?? ?? ]
3.3 Show that ?? ?? is a field. Then find ([?? ]+[?? ])
-?? and (-[?? ])
-?? in ?? ?? .
(2014 : 15 Marks)
Solution:
?? 7
={[0],[1],[2],[3],[4],[5],[6]}
For ?? 7
to be a field, it should satisfy
(i) (?? 7
,+) is an abelian.
(ii) (?? 7
,?? ) is an abelian group, where ?? 7
*
=?? 7
-{0}
(iii) Distributive law.
(i) (?? 7
+)
+ [0] [1] [2] [3] [4] [5] [6]
[0] [0] [1] [2] [3] [4] [5] [6]
[1] [1] [2] [3] [4] [5] [6] [0]
[2] [2] [3] [4] [5] [6] [0] [1]
[3] [3] [4] [5] [6] [0] [1] [2]
[4] [4] [5] [6] [0] [1] [2] [3]
[5] [5] [6] [0] [1] [2] [3] [4]
[6] [6] [0] [1] [2] [3] [4] [5]
? All elements belong to ?? 7
- closure property.
? First row coincide with the top row, then [0] is identity element.
? [0] is in every row & column
? Inverse property satisfied.
([?? ]+[?? ]+[?? ]=[?? +?? ]+[?? ]=[?? +?? +?? ]=[?? ]+[?? +?? ]=[?? ]+([?? ]+[?? ])
? Associative property satisfied.
? Transpose of matrix equal to original matrix.
? Commutative property satisfy.
(ii) (?? 7
*
,?? )
? Every element belong to ?? 7
*
-closure property.
? Top row coincide with first row, Hence [1] is an identity element.
? [1] is in every row & column.
? inverse property satisfied.
? ([?? ]×[?? ]×[?? ]=[(?? ×?? )×[?? ]=[?? ×?? ×?? ]=[?? ]×[?? ×?? ]=[?? ]×([?? ]×[?? ]) .
? Associative property satisfied.
? Transpose of matrix is equal to original matrix.
? Commutative property satisfy.
(iii) Distributivity :
[?? ]×([?? ]×[?? ])?=[?? ]×([?? +?? ])=[?? (?? +?? )]
?=[???? +???? ]=[???? ]+[???? ]=[?? ][?? ]+[?? ][?? ]
([?? ]+[?? ])×[?? ]?=[?? ][?? ]+[?? ][?? ]
Similarly
??????????????????????????????????([5]+[6])
-1
???????? ?=[4]
-1
=[2]
???????????????????????????????????(-[4])
-1
?????????=([3])
-1
=[5]
3.4 Show that the set {?? +???? :?? ?? =?? } , where ?? and ?? are real numbers, is a field
with respect to usual addition and multiplication.
(2014 : 15 Marks)
Solution:
Let ?????????????????????????????????????? ={(?? +???? );?? 3
=1;?? ,?? ?|?? }
(i) Let (?? ,+) be an algebraic structure.
?(?? +???? ),(?? +???? )??? .
(?? +???? )+(?? +???? )?=(?? +?? )+(?? +?? )?? ?=?? +???? -(?? +?? )?|??
? Closure property satisfy.
(ii) Let, (?? +???? ),(?? +???? ),(?? +???? )??? .
[(?? +???? )+(?? +???? )]?=((?? +?? )+(?? +?? )?? )+(?? +???? )
????????????????????????????????????????????????????????????????????????????=(?? +?? +?? )+(?? +?? +?? )w???????????????????????????(i)
(?? +?? +?? )?R
(?? +?? +?? )?|??
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FAQs on Fields - Mathematics Optional Notes for UPSC
| 1. What is the eligibility criteria for the UPSC exam? |  |
Ans. To be eligible for the UPSC exam, candidates must be a citizen of India, have a bachelor's degree from a recognized university, and be between the ages of 21 and 32 years (relaxation in age limit for certain categories).
| 2. How many attempts are allowed for the UPSC exam? | |
Ans. General category candidates can attempt the UPSC exam a maximum of 6 times, while OBC category candidates can attempt it 9 times. There is no limit on the number of attempts for SC/ST category candidates.
| 3. What is the exam pattern for the UPSC exam? | |
Ans. The UPSC exam consists of three stages - Preliminary examination (objective type), Main examination (descriptive type), and Personality Test (Interview). The Preliminary exam has two papers - General Studies and CSAT (Civil Services Aptitude Test).
| 4. How can I prepare for the UPSC exam effectively? | |
Ans. To prepare for the UPSC exam effectively, candidates should create a study schedule, focus on current affairs, practice previous year question papers, and revise regularly. It is also important to stay updated with the syllabus and exam pattern.
| 5. What are the optional subjects available for the UPSC exam? | |
Ans. The UPSC exam offers a wide range of optional subjects for the Main examination, including History, Geography, Public Administration, Sociology, Political Science, and Economics, among others. Candidates can choose one optional subject based on their interest and background.
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About this Document
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Dec 18, 2024 Last updated |
Document Description: Fields for UPSC 2024 is part of Mathematics Optional Notes for UPSC preparation.
The notes and questions for Fields have been prepared according to the UPSC exam syllabus. Information about Fields covers topics
like and Fields Example, for UPSC 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Fields.
Introduction of Fields in English is available as part of our Mathematics Optional Notes for UPSC
for UPSC & Fields in Hindi for Mathematics Optional Notes for UPSC course.
Download more important topics related with notes, lectures and mock test series for UPSC
Exam by signing up for free. UPSC: Fields | Mathematics Optional Notes for UPSC
Description
Full syllabus notes, lecture & questions for Fields | Mathematics Optional Notes for UPSC - UPSC | Plus excerises question with solution to help you revise complete syllabus for Mathematics Optional Notes for UPSC | Best notes, free PDF download
Information about Fields
In this doc you can find the meaning of Fields defined & explained in the simplest way possible. Besides explaining types of
Fields theory, EduRev gives you an ample number of questions to practice Fields tests, examples and also practice UPSC
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Additional Information about Fields for UPSC Preparation
Fields Free PDF Download
The Fields is an invaluable resource that delves deep into the core of the UPSC exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
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allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Fields now and kickstart your journey towards success in the UPSC exam.
Importance of Fields
The importance of Fields cannot be overstated, especially for UPSC aspirants.
This document holds the key to success in the UPSC exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Fields Notes
Fields Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Fields.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
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Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Fields Notes on EduRev are your ultimate resource for success.
Fields UPSC Questions
The "Fields UPSC Questions" guide is a valuable resource for all aspiring students preparing for the
UPSC exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Fields on the App
Students of UPSC can study Fields alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Fields,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Fields is prepared as per the latest UPSC syllabus.
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