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Galois group (Definition)

The Galois group $ \operatorname{Gal}(K/F)$ of a field extension $ K/F$ is the group of all field automorphisms $ \sigma\colon K \to K$ of $ K$ which fix $ F$ (i.e., $ \sigma(x) = x$ for all $ x \in F$). The group operation is given by composition: for two automorphisms $ \sigma_1, \sigma_2 \in \operatorname{Gal}(K/F)$, given by $ \sigma_1\colon K \to K$ and $ \sigma_2\colon K \to K$, the product $ \sigma_1 \cdot \sigma_2 \in \operatorname{Gal}(K/F)$ is the composite of the two maps $ \sigma_1 \circ \sigma_2\colon K \to K$.

The Galois group of a polynomial $ f(x) \in F[x]$ is defined to be the Galois group of the splitting field of $ f(x)$ over $ F$.



"Galois group" is owned by djao.
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See Also: fundamental theorem of Galois theory, infinite Galois theory


Attachments:
Galois group of a cubic polynomial (Topic) by rm50
Galois group of a quartic polynomial (Topic) by rm50
field extension with Galois group $Q_8$ (Example) by rm50
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Cross-references: splitting field, polynomial, maps, composite, product, composition, group operation, fix, automorphisms, field, group, field extension
There are 57 references to this entry.

This is version 4 of Galois group, born on 2002-01-05, modified 2004-07-28.
Object id is 1317, canonical name is GaloisGroup.
Accessed 8388 times total.

Classification:
AMS MSC: 12F10 (Field theory and polynomials :: Field extensions :: Separable extensions, Galois theory)
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