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created: 2023-04-01T23:28
updated: 2023-04-01T23:29
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---
date: 2022-03-25 23:23
modification date: Friday 25th March 2022 23:23:06
title: "trace map"
aliases: [trace map, trace, norm, norm map, additive transfer]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #homotopy/stable-homotopy/equivariant 
- Refs:
	- #todo/add-references
- Links:
	- [norm map](Unsorted/homotopy%20fixed%20points.md)
	- [dualizable](Unsorted/dualizable.md)
	- [Adams isomorphism](Adams%20isomorphism)
	- [Pontrjagin-Thom collapse](Pontrjagin-Thom%20collapse)

---

# Summary

Let $A\in \kalg^\fd$ and let $[m_a]$ be the matrix of $x\mapsto ax$ in a $k\dash$basis.
The **trace** of $a \in A$ is the trace of $m_{a}$. The **norm** of $a$ is the determinant of $m_{a}$.

![](attachments/Pasted%20image%2020220830114611.png)

![](attachments/Pasted%20image%2020221017005924.png)

![](attachments/Pasted%20image%2020230111163313.png)

# trace map

![](attachments/Pasted%20image%2020220417004143.png)

![](attachments/Pasted%20image%2020220328100011.png)

Relation to [separable](Unsorted/separable.md) algebras:
![](attachments/Pasted%20image%2020220417004200.png)


# norm map

![](attachments/Pasted%20image%2020220325232530.png)
![](attachments/Pasted%20image%2020220325232555.png)

# Results

- $a \in A^{\times} \text {if and only if } \mathrm{N}_{A / k}(a) \neq 0$.
- Let $M_a$ be the minimal polynnomial of $a\in A$, then
	  If $D$ is a finite-dimensional $k$-division algebra, then for all $a \in D$,
$$
\chi_{a}=M_{a}^{\operatorname{deg}\left(\chi_{a}\right) \over \operatorname{deg}\left(M_{a}\right)}=M_{a}^{[D: k] \over [k[a]: k]} .
$$
	Thus, $\mathrm{N}_{D / k}(a)=(-1)^{[D: k]} M_{a}(0)^{[D: k] /[k[a]: k]}$.
- $\operatorname{Tr}_{K / k}(a)=\sum_{i} \sigma_{i}(a) \text { and } \mathrm{N}_{K / k}(a)=\prod_{i} \sigma_{i}(a)$.
- For $a \in \mathbf{F}_{q^{n}}$,
$$
\operatorname{Tr}_{\mathbf{F}_{q^{n}} / \mathbf{F}_{q}}(a)=a+a^{q}+\cdots+a^{q^{n-1}} \text { and } \mathbf{N}_{\mathbf{F}_{q^{n}} / \mathbf{F}_{q}}(a)=a \cdot a^{q} \cdots a^{q^{n-1}} .
$$
  

# Examples


![](attachments/Pasted%20image%2020220609124324.png)

![](attachments/Pasted%20image%2020220609124453.png)