An Artin representation $\rho:\textrm{Gal}(\overline{\Q}/\Q)\to \GL(V)$ factors through the Galois group of a finite extension of $\Q$. Thus its image is a finite group $G \subset \GL(V)$ which is canonically isomorphic to the Galois group of a finite extension $K/\Q$ where $K$ is the fixed field of $\ker(\rho)$ and is called the Artin field of $\rho$. We sometimes say that $\Gal(K/\Q)$ is the **image of the Artin representation** $\rho$.

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**Knowl status:**

- Review status: reviewed
- Last edited by Alina Bucur on 2019-05-02 23:29:31

**Referred to by:**

- lmfdb/artin_representations/templates/artin-representation-galois-orbit.html (line 8)
- lmfdb/artin_representations/templates/artin-representation-index.html (line 27)
- lmfdb/artin_representations/templates/artin-representation-index.html (line 69)
- lmfdb/artin_representations/templates/artin-representation-search.html (line 11)
- lmfdb/artin_representations/templates/artin-representation-search.html (line 58)
- lmfdb/artin_representations/templates/artin-representation-show.html (line 8)
- lmfdb/number_fields/templates/nf-show-field.html (line 266)

**History:**(expand/hide all)

- 2019-05-02 23:29:31 by Alina Bucur (Reviewed)
- 2019-05-02 23:27:06 by Alina Bucur
- 2012-06-26 12:50:21 by John Jones

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