In many papers and books, we can find some properties about the numbers represented by quadratic polynomials, especially the norm in quadratic fields. But, it is not easy to see properties about the numbers represented by cubic polynomials in cubic fields. M. A. Mathews[7] found the fact that for some integer , there is infinitely many triples of integers for special cubic equatons. E. J. Barbeau[3] found solutions for special cases of the parameter c of special cubic equations.
In this thesis, we study about the numbers represented by the cubic polynomial in cubic fields. First, we prove Lemma 3.3.4. Second, we Theorem 3.3.5. Third, we prove Theorem 3.3.6. Fourth, we prove Corollary 3.3.8,, 3.3.9, 3.3.10, and 3.3.11. Fifth, we find the fact that there is no integral solution of soem Diophantine equations, see Example 4.1.1, 4.1.2, 4.1.3, and 4.1.4.
