I understand that every degree 2 extension of fields of characteristic not equal to 2 can be obtained by adjoining the square root of a single element. It seems that for extensions in characteristic 2, this statement can be true and it can be false. May I please ask for two specific examples for either case, please? (P.S. I am a beginner in field theory so it would help if some explanations on the two examples can be given.)
Here are a few confusions:
- It seems that I cannot adjoin a square which already exists in the smaller field. I can only adjoin a square root of an element (of the smaller field) which does not exist in the smaller field (By the following proposition from Artin:
Does that mean I cannot adjoin any square root of the elements in $\Bbb Z_2$ because it contains all its square roots?
I still don't know how to formally prove that the field $\Bbb Z_2[x]/<x^2+x+1>$ cannot be obtained by adjoining a square root.
In which case an extension field can be obtained by adjoining a square root? Detailed Explanation needed. Thanks so much.