Definition of normal extension

[emptyboat]

normal extension - an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]. (wikipedia)

I have a question about the 'family of polynomials'
it says the family should be arbitarary large?
if E be an algebraic closure of Q(rational number).
is E a normal extension of Q?

 

[rasmhop]

What is the splitting field of all polynomials with coefficients in Q? If you can show that this is E, then you have shown that E/Q is a normal extension.

 

[emptyboat]

Yes, I got it.
You mean arbitrary large family is admittable.

Thanks a lot !

 

[rasmhop]

Yes, I got it.
You mean arbitrary large family is admittable.

Thanks a lot !

Yes the family can be arbitrarily large.

 

[blue_raver22]

I would like to clarify that the definition of normal extension states that L must be the splitting field of a family of polynomials in K[X]. This family can be any size, as long as L is the splitting field for all the polynomials in K[X]. Therefore, the size of the family does not affect the normality of the extension.

To answer your question, if E is an algebraic closure of Q (rational numbers), then E is indeed a normal extension of Q. This is because E is the splitting field for all polynomials in Q[X], since every polynomial in Q[X] has all its roots in E. Therefore, E meets the criteria for being a normal extension of Q.