Register to reply 
Splitting Fields 
Share this thread: 
#1
Feb1809, 08:31 PM

P: 102

I was thinking about this,
finding the splitting field of x^42 in Q[x] over Q is standard enough... but would much be different is i wanted the splitting field over F_5? (field with 5 elements) would it just be F_5(2^(1/4), i) analogously to the Q case? or do any of the arguments break down? Any thoughts are appreciated, cheers 


#2
Feb1809, 10:21 PM

P: 274

I think it would be just F_5(2^(1/4)), since once you have one fourth root of 2, the others would just be 2*2^(1/4), 4*2^(1/4), and 3*2^(1/4) (since 2^4 = 1 in F_5).



#3
Feb1809, 10:36 PM

P: 102

Very true! However, what does 2^(1/4) mean exactly in this case? i dont think it can be a real number since i dont believe there is an extension from F_5 to R... And since there is no element x in F_5 such that x^4=2... perhaps i am confused? Thanks for you reply! 


#4
Feb1809, 11:11 PM

P: 274

Splitting Fields
No, it wouldn't be an element of R. It would be an element of some algebraic extension of F_5. In this case, since the polynomial x^4  2 is irreducible over F_5, we can take that extension to be the quotient ring F_5[X]/(X^4  2).



Register to reply 
Related Discussions  
Splitting fields again  Calculus & Beyond Homework  16  
Splitting fields  Calculus & Beyond Homework  4  
Splitting fields  Calculus & Beyond Homework  4  
Uniqueness of Splitting Fields  Linear & Abstract Algebra  5  
HNMR splitting  Chemistry  2 