Fractional Part of Algebraic Number

Question

I want to know whether a fractional part of algebraic number is still algebraic (moreover with the same degree). Is the statement true? I was trying to find the minimal polynomial explicitly from the original minimal polynomial but it does not work. For example, I tried to see if the integer part differs by one it can change the polynomial drastically such as $\dfrac{-3\pm \sqrt{5}}{2}$ and $\dfrac{-1\pm \sqrt{5}}{2}$ have minimal polynomials $x^2+3x+1, x^2+x-1$ respectively, which I cannot see the relation between them.

Answer 1

Suppose $a$ has minimal polynomial (over $\mathbb Q$) $p(x)$. Then $a+n$ has minimal polynomial $p(x-n)$, which is clearly of the same degree.

For example, you already know that the minimal polynomial of $a=\frac{-1+\sqrt5}2$ is $x^2+x-1$. That of $a-1=\frac{-3+\sqrt5}2$ is then $(x+1)^2+(x+1)-1=x^2+3x+1$.

Since $n$ can be the integer part of $a$, the result follows.