Question

Let E be a field, F a subfield of E,c∈E, and d∈F. It is a fact that, for any polynomial f(x)∈F[x] c is a root of f(x) if and only if c+d is a root of f(x−d)∈F[x]. Use this fact to prove that c is algebraic over F if and only if c+d is algebraic over F.

Answer 1

Element c is algebraic over F if and only if c+d is algebraic over F. This is proven using the given property that relates the roots of polynomials in F[x] when shifted by a constant in F.

Step-by-step explanation:

To prove that c is algebraic over F if and only if c+d is algebraic over F, one can use the given property: for any polynomial f(x) ∈ F[x], c is a root of f(x) if and only if (c+d) is a root of f(x-d). Being algebraic means there exists a non-zero polynomial such that the element is a root. If c is algebraic over F, there exists a non-zero polynomial f(x) in F[x] where f(c) = 0. By the property, f(c) = 0 implies that f((c+d)-d) = f(c+d) = 0, making c+d a root of the polynomial f(x-d), which belongs to F[x] since d is in F and coefficients are shifted by a constant in F. By definition, this means c+d is algebraic over F. Conversely, if c+d is algebraic over F, the same reasoning applies, showing that there is some polynomial g(x) in F[x] for which g(c+d) = 0, meaning that g(x+d) has c as a root and g(x+d) ∈ F[x]. Hence, c is algebraic over F.