Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

94
ADVANCED ABSTRACT ALGEBRA
by
Then
and
f(a) = 1
a , a ∈ D .
f(a+b) =
f(ab) =
a + b
1
ab =
1
a b
= +
1 1
= f(a) + f(b)
a
b
1
1
= f(a) f(b) .
Therefore f is a ring homomorphism.
Also,
a b
f(a) = f(b) =
1 1
a = b .
It follows therefore that f is a isomorphism. Hence D can be imbedded in F.
Definition. The Quotient field of an integral domain :- By the quotient field K of an integral domain
D is meant the smallest field containing D. Thus a field K is a quotient field of an integral domain D if K
contains D and is itself contained in every field containing D.
For example, field Q of rational numbers is the quotient field of the integral domain Z of integers.
*The quotient field of a finite integral domain coincides with itself.
Definition. Let F be a field. If a subring F 1 of F form a field under the induced compositions of
addition and multiplication, then F 1 is called a subfield of F.
For example, field Q of rational numbers is a subfield of the field R of real numbers. The field R is a
subfield of the field C of complex numbers. Every field is a subfield of itself.
It is clear from the definition that a nonempty set K is a subfield of a field F if
(i)
x, y ∈ K x−y ∈ K
(ii) x∈ K, y ∈ K, y ≠ 0 xy −1 ∈ K .
Characteristic of a field :- Let K be a field and e be the multiplicative identity of K. Then, the
mapping f : Z → K defined by f(n) = ne, n ∈ Z is a ring homomorphism. For,
f(m+n) = (m+n) e
= (me) + (ne)
= f(m) + f(n)
and
f(mn) = (mn)e
= (me)(ne)
= f(m)f(n) .
Let A be the kernel of this homorphism. Then
A = {n | f(n) = 0 }
= {n | ne = 0} (i)