Associated Quadratic Field
The discriminant of a cubic field K can be written uniquely as df2 where d is a fundamental discriminant. Then, K is cyclic if, and only if, d = 1, in which case the only subfield of K is Q itself. If d ≠ 1, then the Galois closure N of K contains a unique quadratic field k whose discriminant is d (in the case d = 1, the subfield Q is sometimes considered as the "degenerate" quadratic field of discriminant 1). The conductor of N over k is f, and f2 is the relative discriminant of N over k. The discriminant of N is d3f4.
The field K is a pure cubic field if, and only if, d = −3. This is the case for which the quadratic field contained in the Galois closure of K is the cyclotomic field of cube roots of unity.
Read more about this topic: Cubic Field
Famous quotes containing the word field:
“Beat! beat! drums!—blow! bugles! blow!
Through the windows—through doors—burst like a ruthless force,
Into the solemn church, and scatter the congregation;
Into the school where the scholar is studying;
Leave not the bridegroom quiet—no happiness must he have now with his bride;
Nor the peaceful farmer any peace, plough his field or gathering his
grain;
So fierce you whirr and pound, you drums—so shrill you bugles blow.”
—Walt Whitman (1819–1892)