Question about kth root of a reduced ring element.

Question

Let $n > 1$ be a positive integer. Let $k > 1$ be a positive integer. Define the reduced polynomial rings $f_n = \Bbb R[X_n]/(1+(X_n)^{n})$

How do we know if $(X_n)^{1/k}$ is an element of $f_n$ ? Possibly trivially related : How do we know if $(-1)^{1/k}$ is an element of $f_n$ ?

For instance I wonder if $(X_5)^{1/3}$ is an element of $f_5$ and if $(-1)^{1/7}$ is an element of $f_7$.

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Im new to ring theory so I start with this simple case but ofcourse this can be generalized by replacing the reduced rings $f_n$ with $g_n = \Bbb R[X_n]/(G_n(X_n))$ where $G_n$ is a polynomial of degree $n$ (that keeps the ring reduced).

( Btw I also wonder about subrings but I do not have a clear and precise question for it).

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A naive conjecture would probably be $(X_n)^{1/k}$ is not an element of $f_n$ iff $gcd(n,k)=1$ , but I do not know how to disprove it.

I would like to see patterns and theorems. I assume primes are important here.

mick

Answer 1

I figured it out partially.

The quotient $\Bbb{R}[X_n]/(G_n(X_n))$ is reduced if and only if $G_n(X_n)$ is a product of distinct irreducible factors. Suppose $P_1,\ldots,P_m\in\Bbb{R}[X_n]$ are distinct irreducible polynomials such that $$G_n(X_n)=\prod_{i=1}^mP_i(X_n),$$ then by the Chinese remainder theorem we have $$\Bbb{R}[X_n]/(G_n(X_n))\cong\prod_{i=1}^m\Bbb{R}[X_n]/(P_i(X_n))\cong\Bbb{R}^r\times\Bbb{C}^s,$$

Since a reduced polynomial ring is thus isomorphic to a copy of reals and complexes , whenever any of those reals has the value -1 then we cannot take the square root.

SO if a ring element $A$ is iso to $(a,b)$ where $a$ is restricted to reals and $b$ to complex then

$\sqrt(a,b)$ is not possible and the element $\sqrt A$ does not exist.

Answer 2

Yes, the gcd is a good idea. Then progress to the extended gcd or Bezout identity to find modular inverses, i.e., if $gcd(k,2n)=1$, then there is some $m$ with $km\equiv 1\pmod {2n}$ and thus $$ X_n\equiv X_n^{km}=(X_n^m)^k\pmod{1+X_n^{n}} $$ so that $X_n^m$ is the $k$-th root of $X_n$.

Now one would have to explore the other cases, especially the even degree roots.