|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
proof of norm and trace of algebraic number
|
(Proof)
|
|
Theorem 1 Let $K$ be a number field and $\alpha \in K$ . The norm $N(\alpha)$ and the trace $T(\alpha)$ of $\alpha$ in the field extension $K/\mathbb{Q}$ both are rational numbers and especially rational integers in the case
$\alpha$ is an algebraic integer. If $\beta$ is another element of $K$ , then $N(\alpha\beta) = N(\alpha)N(\beta)$ and $T(\alpha+\beta) = T(\alpha)+T(\beta)$ . If $[K\!:\!\mathbb{Q}] = n$ and $a\in\mathbb{Q}$ , then $N(a) = a^n$ and $T(a) = na$ .
Before proving this theorem, a lemma will be stated and proven.
Lemma Let $K$ be a number field with $[K\!:\!\mathbb{Q}]=n$ , $\alpha \in K$ such that $[\mathbb{Q}(\alpha)\!:\!\mathbb{Q}]=d$ , and $N^*(\alpha)$ and $T^*(\alpha)$ denote the absolute norm and absolute trace of $\alpha$ , respectively. Then $d$ divides $n$ , $\displaystyle N(\alpha)=(N^*(\alpha))^{\frac{n}{d}},$ and $\displaystyle T(\alpha)=\frac{n}{d}T^*(\alpha)$ .
Proof. Note that $d$ divides $n$ because $n=[K\!:\!\mathbb{Q}]=[K\!:\!\mathbb{Q}(\alpha)][\mathbb{Q}(\alpha)\!:\!\mathbb{Q}]=[K\!:\!\mathbb{Q}(\alpha)]d$ .
Note also that each of the $d$ embeddings of $\mathbb{Q}(\alpha)$ into $\mathbb{C}$ extends to exactly $\displaystyle \frac{n}{d}$ embeddings of $K$ into $\mathbb{C}$ . Thus,
and
Now, the above theorem will be proven.
Proof of theorem 1. Let $f(x) \in \mathbb{Q}[x]$ be the minimal polynomial for $\alpha$ over $\mathbb{Q}$ . Then $\operatorname{deg} f=d$ , where $d$ is as in the previous lemma. Note that $|N^*(\alpha)|$ is equal to the absolute value of the constant term of $f$ and that $T^*(\alpha)$ is equal to the opposite of the coefficient of $x^{d-1}$ of $f$ . Thus, $N^*(\alpha), T^*(\alpha) \in \mathbb{Q}$ . Therefore, $\displaystyle N(\alpha)=(N^*(\alpha))^{\frac{n}{d}} \in \mathbb{Q}$ and $\displaystyle T(\alpha)=\frac{n}{d}T^*(\alpha) \in \mathbb{Q}$ . Moreover, if $\alpha$ is an algebraic integer, then $f(x) \in \mathbb{Z}[x]$ , $N^*(\alpha), T^*(\alpha) \in \mathbb{Z}$ , $\displaystyle N(\alpha)=(N^*(\alpha))^{\frac{n}{d}} \in \mathbb{Z}$ , and $\displaystyle T(\alpha)=\frac{n}{d}T^*(\alpha) \in \mathbb{Z}$ .
If $a \in \mathbb{Q}$ , then $d=1$ , $N(a)=(N^*(a))^n=a^n$ , and $T(a)=nT^*(a)=na$ .
Finally, if $\alpha, \beta \in K$ , then
and
$\qedsymbol$
Theorem 2 An algebraic integer $\varepsilon$ is a unit if and only if its absolute norm $N^*(\varepsilon)=\pm 1,$ . Thus, the constant term in the minimal polynomial of an algebraic unit is always $\pm 1$ .
Proof. Let $K=\mathbb{Q}(\varepsilon)$ . Since $\varepsilon$ is an algebraic integer, $d=[K\!:\!\mathbb{Q}]$ is finite. Let $\mathcal{O}_K$ denote the ring of integers of $K$ .
If $N^*(\varepsilon) = \pm 1$ , then let $f(x) \in \mathbb{Z}[x]$ be the minimal polynomial of $\varepsilon$ over $\mathbb{Q}$ . Let $a_1, \cdots , a_{d-1} \in \mathbb{Z}$ such that $\displaystyle f(x)=x^d+\sum_{j=1}^{d-1} a_j x^j \pm 1$ . Then $\displaystyle 0=f(\varepsilon)=\varepsilon^d+\sum_{j=1}^{d-1} a_j \varepsilon^j \pm 1$ . Thus, $\displaystyle \varepsilon \left( \varepsilon^{d-1}+\sum_{j=1}^{d-1} a_j \varepsilon^{j-1} \right) = \pm 1$ . Since $\displaystyle \varepsilon^{d-1}+\sum_{j=1}^{d-1} a_j \varepsilon^{j-1} \in \mathcal{O}_K$ , it follows that $\varepsilon$ is a unit in $\mathcal{O}_K$ .
Conversely, let $\varepsilon$ be a unit in $\mathcal{O}_K$ . Let $\upsilon \in \mathcal{O}_K$ with $\varepsilon \upsilon = 1$ . Since $N^*(\varepsilon) N^*(\upsilon) = N^*(\varepsilon \upsilon)=N^*(1)=1$ and $N^*(\varepsilon), N^*(\upsilon) \in \mathbb{Z}$ , it follows that $N^*(\varepsilon) = \pm 1$ . 
- 1
- Marcus, Daniel A. Number Fields. New York: Springer-Verlag, 1977.
|
"proof of norm and trace of algebraic number" is owned by Wkbj79.
|
|
(view preamble | get metadata)
Cross-references: conversely, ring of integers, finite, algebraic unit, unit, coefficient, opposite, constant term, absolute value, minimal polynomial, proof, embeddings, divides, theorem, algebraic integer, rational integers, rational numbers, field extension, trace, norm, number field
This is version 24 of proof of norm and trace of algebraic number, born on 2006-06-10, modified 2008-02-26.
Object id is 7998, canonical name is ProofOfNormAndTraceOfAlgebraicNumber.
Accessed 2024 times total.
Classification:
| AMS MSC: | 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
