To define the concept of a subfield of a field and prove the stated property regarding subfields of a field.
A subfield of a field L is a subset K of L , which is also a field, with field structure inherited from L.
Let L be a field.
Definition:
A subset
Examples:
1)
2)
Let L be a field.
Let
Claim:
Proof:
So, K is a commutive ring with 1
ANSWER: proved that the intersection of any collection of subfields of a field L is indeed a field, (in fact , a subfield of L)
Also,
rArr
rArr
Thus, K is a field, in fact, K is a subfield of L
Consider the incomplete character table for a group given below:
All the conjugacy classes are there.
1) What is the order of the group?
2) How many characters are missing?
3) Find the missing character and complete the table.
4) Find the order of the Kemel of the missing character.