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properties of Q(ϑ)-conjugates∗
pahio†
2013-03-22 3:21:41
Lemma. Let α1 , α2 , . . . , αs be algebraic numbers belonging to the number
(j)
field Q(ϑ) of degree n and αi their Q(ϑ)-conjugates. If P (x1 , x2 , . . . , xs ) is a
polynomial with rational coefficients and if
P (α1 , α2 , . . . , αs ) = 0,
then also
(j)
(j)
P (α1 , α2 , . . . , αs(j) ) = 0
for each j = 1, 2, . . . , n. In the special case of two elements α and β of Q(ϑ)
one may infer the formulae
(αβ)(j) = α(j) β (j) ,
(α+β)(j) = α(j) +β (j) .
(1)
The lemma implies easily the following theorems.
Theorem 1. All conjugate fields of Q(ϑ) are isomorphic.
Theorem 2. The norm and the trace in the field Q(ϑ) satisfy
N(αβ) = N(α)N(β),
S(α+β) = S(α)+S(β).
Cf. the entry norm and trace of algebraic number.
∗ hPropertiesOfmathbbQvarthetaconjugatesi
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