Find the subfield K of C generated by X= {1,i}

[futurebird]

Find the subfield K of C generated by X= {1,i}


It says "Since K is closed under the arithmetical operations, it must
contain all complex numbers of the form p + qi, where p,q are in Q
(the rationals)."

But, why can't p and q be real? I don't understand why they must be rational.

Any idea?

 

[Office_Shredder]

The subfield generated by 1 and i is going to be the smallest field containing 1 and i. If K is the smallest possible field containing 1 and i, it must contain every rational number, and hence every number of the form p+qi where p and q are rational just by noting K is closed under addition and multiplication. Just using field operations cannot get you, for example, the square root of 2 to be required in the field.

Note that the smallest subfield containing two elements is not always just rational linear combinations of those elements, but it happens to be here (and that's something either proven in the book or that you should try to prove yourself). But either way, it must contain the linear combinations of the generators because a field is closed under addition, multiplication and inverse taking