Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

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ADVANCED ABSTRACT ALGEBRA
4. If K is a finite extension of F, every element of K is algebraic over F.
Example 2.
e j e j e j e j
Q 2, 3 : Q = Q 2,
3 : Q 2 Q 2 : Q = 4
e j e j e j
Put Q 2 = L , Q 2 , 3 = L 3 .
e j e j e j
bg
Then Q 2 , 3 : Q 2 = L 3 : L = 2,
the degree of independent polynomial p x
e j rre j
i.e. L 3 : L = deg I L, 3 = 2 .
e j rre j
e j
2
= x − 3 over L = Q 2 .
Q 2 : Q = deg I Q, 2 = 2 (from example 1)
Hence the result.
5. If p(x) is an irreducible polynomial of degree n in F[x], then F x F c
p x ≅
p(x). By (2), F(c) is of degree n over F.
If
Example 3.
are roots of the same inducible polynomial p(x) over F, then F α ≅ F β .
bg= + +
2
We construct a field of four elements. p x x x
bg bg , where e is a root of
bg bg
di
di
1 is ireducible in Z2 x , as p 0 ≠ 0, p 1 ≠ 0 .
α
c 2
Hence Z x 2
Z2
c
pbg x ≅ , where e is a root of p(x).
i.e.
Now elements of Z 2
(e) are {0, 1, c, c+1} which is illustrated from the following tables:
+
2
0 1 c c + 1
0 0 1 c c + 1
1 1 0 1+
c c
c c 1+
c 0 1
c + 1 c + 1 c 1 0
•
2
1 c c + 1
1 1 c c + 1
c c c + 1 1
c + 1 c + 1 1 c