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=Main Discussion= | =Main Discussion= | ||
| − | + | ==== Galois Group ==== | |
| + | Now that groups and fields have been described, it is time to define the Galois group. | ||
| + | For starters, define a group G. Referring back to field extensions, if there exists an extension F of Q, then there exists a grouping of automorphisms of Q onto F. Let the group G be the container of these automorphisms. In general, this basic definition is referred to as the Galois group of the field extension. However, if the field F is actually the splitting field of a polynomial, then it can be called the Galois group of that polynomial. | ||
| + | If the Galois group is a grouping of automorphisms of a field, then how can one know it is a group? What is its operation? A Galois group makes use of function composition as its operation, | ||
| + | |||
| + | So, what's the purpose of such an abstractly defined structure? The answer to this question involves what a Galois group is capable of doing. For instance, if a Galois group is found for a polynomial p(x), and one proves that this Galois group is soluble, then the polynomial has radical roots. | ||
[[ Walther MA271 Fall2020 topic1|Back to Walther MA271 Fall2020 topic1]] | [[ Walther MA271 Fall2020 topic1|Back to Walther MA271 Fall2020 topic1]] | ||
[[Category:MA271Fall2020Walther]] | [[Category:MA271Fall2020Walther]] | ||
Revision as of 15:12, 6 December 2020
Main Discussion
Galois Group
Now that groups and fields have been described, it is time to define the Galois group.
For starters, define a group G. Referring back to field extensions, if there exists an extension F of Q, then there exists a grouping of automorphisms of Q onto F. Let the group G be the container of these automorphisms. In general, this basic definition is referred to as the Galois group of the field extension. However, if the field F is actually the splitting field of a polynomial, then it can be called the Galois group of that polynomial.
If the Galois group is a grouping of automorphisms of a field, then how can one know it is a group? What is its operation? A Galois group makes use of function composition as its operation,
So, what's the purpose of such an abstractly defined structure? The answer to this question involves what a Galois group is capable of doing. For instance, if a Galois group is found for a polynomial p(x), and one proves that this Galois group is soluble, then the polynomial has radical roots. Back to Walther MA271 Fall2020 topic1
