My field of research is coding theory and I am working on cyclic codes. During my research, I tackled an algebraic problem. After some simple definitions, I asked my question. I will appreciate any helpful answer and comment.

Let $F_{2^{2m}}$ denote the finite field of ${2^{2m}}$ elements, where $m$ is a positive integer. Let $F_{2^{2m}}[x]$ denote the polynomial ring in indeterminate $x$ with coefficients from $F_{2^{2m}}$.

Suppose that $f(x)$ is a polynomial in $F_{2^{2m}}[x]$ and $f(x)=f_0+f_1x+\cdots+f_kx^k$. We define the conjugate polynomial of $f(x)$ over $F_{2^{2m}}$ as follows:

$\overline{f(x)}={f_0}^{2^m}+f_1^{2^m}x+\cdots +f_k^{2^m}x^k.$

In particular, if a polynomial is equal to its conjugate polynomial over $F_{2^{2m}}$, then it is called self-conjugate polynomial.

Let $n$ be an odd positive integer. Since $gcd(n,2^{2m})=1$, the polynomial $x^n+1$ can be factorized into distinct irreducible polynomials over $F_{2^{2m}}$.

It is obvious that for any monic irreducible polynomial dividing $x^n+1$ over $F_{2^{2m}}$, its conjugate polynomial is also a monic irreducible polynomial dividing $x^n+1$ over $F_{2^{2m}}$.

For example, let $\omega$ be a primitive element of $F_4$. The factorization of $x^5+1$ over $F_4$ is

$x^5+1=(x+1)(x^2+\omega x+1)(x^2+\omega^2x+1)$

It is obvious that $x+1$ is a self-conjugate polynomial and $x^2+\omega x+1$ is the conjugate polynomial of $x^2+\omega^2x+1$ over $F_4$.

For another example, let $\omega$ be a primitive element of $F_{16}$. The factorization of $x^{11}+1$ over $F_{16}$ is

$x^{11}+1=(x+1)(x^5+\omega^5 x^4+x^3+x^2+\omega^{10}x+1)(x^5+\omega^{10} x^4+x^3+x^2+\omega^{5}x+1)$

It is obvious that $x+1$, $x^5+\omega^5 x^4+x^3+x^2+\omega^{10}x+1$ and $x^5+\omega^{10} x^4+x^3+x^2+\omega^{5}x+1$ are self-conjugate polynomials over $F_{16}$.

Because of my researches I think that if $f(x)$ is a self-conjugate monic irreducible polynomial dividing $x^n+1$ over $F_{2^{2m}}$, then the degree of $f(x)$ is odd, but I could not prove it.

Is this conjecture true in general? If the answer is no, please give me an example?