Algebraic Extension

An extension field of field is an algebraic extension of if every element in is algebraic over .
If an extension field of a field is of finite dimension as a vector space over , then is a finite extension of degree over . The degree of over is denoted
(Fraleigh 31.3) A finite extension field of a field is an algebraic extension of .
- Proof: Take . The set forms a linearly dependent set. Their linear combination gives a nonzero polynomial that evaluates to . Therefore is algebraic over .
(Fraleigh 31.4) If is a finite extension field of and is a finite extension field of , then is a finite extension of and

¹

(Fraleigh 31.6) By induction if we have fields such that is a finite extension of , then is a finite extension of and
- Intuition: If is a basis for over is a basis for over . Then, is a basis for over .
- (Fraleigh 31.7) If is an extension field of , is algebraic over and , then divides

(Fraleigh 31.11) Let be an algebraic extension of a field . Then there exists a finite number of elements such that if and only if is a finite-dimensional vector space over . That is, if and only if is a finite extension of
- Proof: In the forward case, is algebraic over . is algebraic over and so on. Now get
  
  Which is finite.
  
  Conversely, if is a finite algebraic extension of , we can keep adding elements where . We do it such that This construction gives us
(Fraleigh 31.23) If is a finite extension of field and is a prime number, then is a simple extension of . Thus
- Proof: We have for any Note that the factors of a prime are and . Clearly since , Therefore since . Consequently, and thus

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