quadratic extension
Let be a field and be its algebraic closure. Suppose that . A quadratic extension over is a field such that for some , where .
If , we often write . Every element of can be written as , for some . This representation is unique and we see that is a basis for the vector space over . In fact, we have the following
Proposition. If the characteristic of is not , then is a quadratic extension over iff (as a vector space) over .
Proof.
In the proposition above, the assumption that can not be dropped. If fact, quadratic extensions of do not exist, for if , then .
For the rest of the discussion, we assume that .
Pick any element in . Then and . So is a root of the irreducible polynomial in . If we define to be , then is the other root of , clearly also in . This implies that the minimal polynomial of every element in has degree at most 2, and splits into linear factors in .
Since , are two distinct roots of . This shows that is separable over .
Now, let be any irreducible polynomial over which has a root in . Then the minimal polynomial of in must divide . But because is irreducible, . This shows that is normal over . Since is both separable and normal over , it is a Galois extension over .
Let be an automorphism of fixing . Then is easily seen to be a root of the minimal polynomial of . As a result, either on or is the involution that maps each to . We have just proved
Theorem. Suppose . Any quadratic extension of is Galois over , whose Galois group is isomorphic to .
Remark. A quadratic extension (of a field) is also known in the literature as a -extension, a special case of a p-extension, when .
Title | quadratic extension |
---|---|
Canonical name | QuadraticExtension |
Date of creation | 2013-03-22 15:42:34 |
Last modified on | 2013-03-22 15:42:34 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 20 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 12F05 |
Classification | msc 12F10 |
Synonym | -extension |
Related topic | PExtension |