Math Help - normal extension of the rationals - quick questions

Just trying to get my head around normal extensions.

True or False?

is a normal extension of

is a normal extension of

I would say for the first one, true, because it is the splitting field of .

For the second one, false, because is irreducible by Eisenstein, and has only one zero > 0 in

Could anyone please tell me if these are OK?

Thanks in advance!

Looks good to me.
You can tell the second is not normal because the other two roots are imaginary and . But yeah I think you got it.