Abstract Algebra and Algebraic Number Theory

We define norm,N E/F (α) , trace, T E/F (α) and characteristic polynomial,char E/F (x),
of α, relative to extension E/F , as follows
N E/F (α) = det m(α)
T E/F (α) = trace m(α) and char E/F (α)(x) = det [xI−A(α)]
Proposition 4.2.1. char E/F (α)(x) = [min α,F (x)] r , where r = [E : F (α)].
Corollary. Let [E : F ] = n and [F (α) : F ] = d. Let α 1 , α 2 , ...., α d be the
roots of min α,F (x), counting multiplicity, in a splitting field. Then
N(α) = ( d ∏
i=1
α i
)
, T (α) =
(n
d
d∑ )
α i ,
i=1
{ ∏
d
char(α)(x) = (x − α i ) } n d
i=1
Proof. Result follows from the above theorem and from the fact that
.
char(α)(x) = x n − T (α)x n−1 + ... + (−1) n N(α)
Proposition 4.2.2. Let E/F be a separable extension of degree n, let
σ 1 , σ 2 , ..., σ n be the distinct F-embedding of E into an algebraic closure of
E, or equally well into a normal extension L of F containing E. Then
N E/F (α) =
n∏
σ i (α), T E/F (α) =
i=0
char E/F (α)(x) =
n∑
σ i (α)
i=0
n∏
(x − σ i (α))
Proposition 4.2.3. Let us consider AKLB setup. Let α ∈ B, then the
coefficient of min α,F (x) and char E/F (α)(x) are integral over A, In particular
T L/K (α) and N L/K (α) are integral over A. If A is integrally closed then
coefficient belongs to A.
Corollary. An algebraic integer a ∈ Q must in fact belong to Z.
Proposition 4.2.4. In AKLB setup, let α ∈ L, then there is a non zero
element a ∈ A and β ∈ B such that α = β a
, i.e. L is a fraction field of B.
Proposition 4.2.5. In AKLB setup, there is a basis of L/K consisting
entirely the elements of B.
i=0
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