quality:correction
Modern approach by field theory

In the modern approach, one starts with a field extension L / K (read " L over K "), and examines the group of automorphisms of L that fix K. See the article on Galois groups for further explanation and examples.

The connection between the two approaches is as follows. The coefficients of the polynomial in question should be chosen from the base field K. The top field L should be the field obtained by adjoining the roots of the polynomial in question to the base field K. Any permutation of the roots which respects algebraic equations as described above gives rise to an automorphism of L / K, and vice versa.

In the first example above, we were studying the extension Q (√ 3)/ Q, where Q is the field of rational numbers, and Q (√ 3) is the field obtained from Q by adjoining √ 3. In the second example, we were studying the extension Q (A, B, C, D)/ Q.

There are several advantages to the modern approach over the permutation group approach.