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norm and trace of algebraic number∗
pahio†
2013-03-21 19:18:12
Theorem 1. Let K be an algebraic number field and α an element of K. The
norm N(α) and the trace S(α) of α in the field extension K/Q both are rational
numbers and especially rational integers in the case α is an algebraic integer.
If β is another element of K, then
N(αβ) = N(α)N(β),
S(α+β) = S(α)+S(β),
(1)
i.e. the norm is multiplicative and the trace additive. If [K : Q] = n and a ∈ Q,
then
N(a) = an , S(a) = na.
Remarks
1. The notions norm and trace were originally introduced in German language as “die Norm” and “die Spur”. Therefore in German and many other
literature the symbol of trace is S, Sp or sp. Nowadays the symbols T and Tr
are common.
2. The norm and trace of an algebraic number α in the field extension
Q(α)/Q, i.e. the product and sum of all algebraic conjugates of α, are called
the absolute norm and the absolute trace of α. Formulae like (1) concerning the
absolute norms and traces are not sensible.
Theorem 2. An algebraic integer ε is a unit if and only if
N(ε) = ±1,
i.e. iff the absolute norm of ε is a rational unit. Thus the constant term in the
minimal polynomial of an algebraic unit is always ±1.
√
Example. The minimal polynomial √
of the number 2 + 3, which is the
fundamental unit of the quadratic field Q( 3), is x2 −4x+1.
∗ hNormAndTraceOfAlgebraicNumberi
created: h2013-03-21i by: hpahioi version: h37125i
Privacy setting: h1i hTheoremi h11R04i
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