About a subset of $\mathbb Q[x]$ of polynomials $f$ such that $f(n)=f(-n)$ for every $n$ in $\mathbb N$

Question

Let $A=\{f \in \mathbb Q[x] : f(n)=f(-n)$, for every $n \in \mathbb N\}$.

Show that

1. $A$ is a subring of $\mathbb Q[x]$.
2. $A$ is a Euclidean Domain.
3. For every $f \in A$ we have $f(r)=f(-r)$, for every $r \in \mathbb Q$.

I think I managed to prove 1 and 2, this is how I proceeded:

1

We have that $A \neq \emptyset$, since the constant polynomial $1$ is such that $1(n)=1=1(-n)$, for every $n \in \mathbb N$.

For every $f$, $g \in A$, for every $n \in \mathbb N$ we have:

$(f-g)(n)=f(n)-g(n)=f(-n)-g(-n)=(f-g)(-n)$

and

$(fg)(n)=f(n)g(n)=f(-n)g(-n)=(fg)(-n)$

These mean that both $f-g$ and $fg$ are in $A$. This proves that $A$ is a subring of $\mathbb Q[x]$.

2

For every $f \in A$, we have:

$f=a_nx^n+\cdots+a_0$

where, for each $i \in \{0,\ldots,n\}$, $a_i \in \mathbb Q$.

We have that, for each $q \in \mathbb Q$ seen as a polynomial in $\mathbb Q[x]$, $q(n)=q=q(-n)$, for every $n \in \mathbb N$. This implies that $\mathbb Q \subseteq A$.

If every rational number is in $A$, this implies that every polynomial $f$ in $A$ has a leading coefficient $a_n$ that is a unit in $A$.

I consider the function $deg : A \to \mathbb N$, which assigns to every $f \in A$ its degree. This function serves as a Euclidean evaluation, that means it satisfies:

For every $f$, $g \in A$, with $g \neq 0$, there exist $q$, $r \in A$ such that:

• $f=gq+r$
• $r=0$ or $deg(r)<deg(g)$

This evaluation, together with the fact that every polynomial in $A$ has a unit as leading coefficient, ensure that $A$ is a Euclidean domain.

Now, my questions are:

• Are my solutions correct?
• Can you help me with point 3? There should be some way to exploit the fact that every pair of polynomials can go through Euclidean division but I can't see where to begin.

Answer 1

For $(3)$, if $f \in A$, then the polynomial $f(x)-f(-x)$ has all natural numbers as roots, of which there are infinitely many, and so must be the zero polynomial. This means that $f(r)$ must in fact be equal to $f(-r)$ for all rational numbers $r$. In other words, a polynomial with rational coefficients that becomes an even function when restricted to the integers must in fact be an even function on all rational numbers.

Knowing that $(3)$ is true together with the fact that a polynomial is an even function if and only if all of its nonzero terms have an even degree will then help answer $(2)$.

Namely, consider the map $\varphi:\mathbb{Q}[x] \to A$ that sends a polynomial $f$ to the polynomial obtained from $f$ by doubling the degrees of all its terms, i.e. $f(x^2)$. Then, the aforementioned fact about even functions tells us that $\varphi$ is an isomorphism, and so $A$, being isomorphic to the Euclidean domain $\mathbb{Q}[x]$, must itself be a Euclidean domain.

Answer 2

If $f(x)\in A$, let $g(x)=f(x)-f(-x)$. Then $(\forall n\in\Bbb N):g(n)=0$. Since $g(x)$ has infinitely many zeros, it is the null polynomial. Therefore, $(\forall r\in\Bbb Q):f(r)=f(-r)$.