To find a number that is algebraic with degree 3 over Z_3, one must identify an appropriate extension field of Z_3. The algebraic closure of Z_3 is suggested as a potential solution, although its constructiveness is questioned. It is established that for any extension field E and number α, there exists a mapping π from the polynomial ring (\mathbb{Z} / 3\mathbb{Z})[t] to E such that π(t) equals α.
PREREQUISITESMathematicians, particularly those focused on abstract algebra, students studying field theory, and anyone interested in the properties of algebraic numbers over finite fields.
How about the algebraic closure of Z_3? Or is that too nonconstructuve?ehrenfest said:Homework Statement
I want to find a number that is algebraic with degree 3 over Z_3. To do this, I need to find an extension field of Z_3. Q,R, and Z_p (p greater than 3) definitely will not work because they have different algebra. Anyone?