Normal extensions are a key concept in Galois Theory, bridging the gap between field extensions and polynomial splitting. They're special because every irreducible polynomial with a root in the extension splits completely there.
Understanding normal extensions is crucial for grasping the . They're closely tied to splitting fields and help us analyze the structure of field extensions and their automorphisms.
Normal Extensions
Definition and Properties
Top images from around the web for Definition and Properties
galois theory - reducing depressed quartic to cubic polynomial - Mathematics Stack Exchange View original
Is this image relevant?
abstract algebra - Understanding a Proof in Galois Theory - Mathematics Stack Exchange View original
Is this image relevant?
GaloisGroupProperties | Wolfram Function Repository View original
Is this image relevant?
galois theory - reducing depressed quartic to cubic polynomial - Mathematics Stack Exchange View original
Is this image relevant?
abstract algebra - Understanding a Proof in Galois Theory - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
Top images from around the web for Definition and Properties
galois theory - reducing depressed quartic to cubic polynomial - Mathematics Stack Exchange View original
Is this image relevant?
abstract algebra - Understanding a Proof in Galois Theory - Mathematics Stack Exchange View original
Is this image relevant?
GaloisGroupProperties | Wolfram Function Repository View original
Is this image relevant?
galois theory - reducing depressed quartic to cubic polynomial - Mathematics Stack Exchange View original
Is this image relevant?
abstract algebra - Understanding a Proof in Galois Theory - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
An extension L/K is normal if every irreducible polynomial in K[x] that has a root in L splits completely in L[x]
Example: The extension Q(32)/Q is normal because the minimal polynomial x3−2 splits completely in Q(32)[x]
A is algebraic, meaning every element of L is a root of some polynomial in K[x]
The composition of normal extensions is normal
If L/K and M/L are normal, then M/K is normal
Example: If Q(2)/Q and Q(2,i)/Q(2) are normal, then Q(2,i)/Q is normal
Every finite extension of a finite field is normal
Example: The extension F8/F2 is normal
Equivalence of Definitions
Prove that an extension L/K is normal if and only if L is the of a family of polynomials in K[x]
Show that an extension L/K is normal if and only if every K-embedding of L into an of K maps L onto itself
Example: For the normal extension Q(2)/Q, any Q-embedding of Q(2) into C maps Q(2) onto itself
Demonstrate that an algebraic extension L/K is normal if and only if the of the group of K-automorphisms of L is precisely K
Example: For the normal extension Q(2)/Q, the fixed field of the group of Q-automorphisms of Q(2) is precisely Q
Normal Extensions and Splitting Fields
Characterization
A finite extension L/K is normal if and only if L is the splitting field of some polynomial f(x) in K[x]
Example: The extension Q(2,i)/Q is normal because it is the splitting field of the polynomial x4−2
If L/K is normal, then L is the splitting field of the minimal polynomial of any element α in L over K
Every splitting field is a normal extension
Example: The splitting field of x3−2 over Q is Q(32,ω), which is a normal extension
Fundamental Theorem of Galois Theory
Use the Fundamental Theorem of Galois Theory to check if the corresponding Galois group acts transitively on the roots of the minimal polynomial of a primitive element of L over K
Example: For the normal extension Q(2,i)/Q, the Galois group acts transitively on the roots of the minimal polynomial x4−2 of the primitive element 2+i
Definitions of Normal Extensions
Refer to the content under "Definition and Properties" and "Equivalence of Definitions" in the "Normal Extensions" section above.
Identifying Normal Extensions
Checking Irreducible Polynomials
Check if every irreducible polynomial in K[x] that has a root in L splits completely in L[x]
Example: To show that Q(2,i)/Q is normal, check that every irreducible polynomial in Q[x] with a root in Q(2,i) splits completely in Q(2,i)[x]
Verifying Splitting Fields
Determine if L is the splitting field of some polynomial f(x) in K[x]
Example: To show that Q(32)/Q is normal, verify that it is the splitting field of the polynomial x3−2
Examining Embeddings and Automorphisms
Verify if every K-embedding of L into an algebraic closure of K maps L onto itself
Examine if the fixed field of the group of K-automorphisms of L is precisely K, assuming L/K is algebraic
Example: To show that Q(2)/Q is normal, verify that every Q-embedding of Q(2) into C maps Q(2) onto itself and that the fixed field of the group of Q-automorphisms of Q(2) is precisely Q