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Hello gentle people. Let's see an example.

 f=x^2+4
 g=x^4+16
 F=NumberField(f,'a')

The extension field of F by g cannot obtained by F.extension(g(s),'s') although we input s=var('s') before. Because g is reducible in F. ( g=(x^2-4I)(x^2+4I) )

Can I check irreducibility of a polynomial in a number field?

Is there any method which makes a number field by given polynomial?

Thanks in advance.

Hello gentle people. Let's see an example.

 f=x^2+4
 g=x^4+16
 F=NumberField(f,'a')

The extension field of F by g cannot obtained by F.extension(g(s),'s') although we input s=var('s') before. Because g is reducible in F. ( g=(x^2-4I)(x^2+4I) g=(x^2-4I)(x^2+4I) )

Can I check irreducibility of a polynomial in a number field?

Is there any method which makes a number field by given polynomial?

Thanks in advance.