an isomorphism from $\mathbb{F}[X]$ to $\mathbb{F}[X]\oplus \mathbb{F}[X]$ Exhibit one isomorphism $\phi:\mathbb{F}[X]\rightarrow\mathbb{F}[X]\oplus \mathbb{F}[X]$, where $\mathbb{F}$ is a field and $\mathbb{F}[X]$ is the vector space of all polynomials with coefficients in $\mathbb{F}$. 
Sorry, but I have no ideas.
 A: Hint: Try the inverse of the mapping (prove that this is an $F$-linear bijection) 
$$\psi:F[x]\oplus F[x]\rightarrow F[x],\quad (a(x),b(x))\mapsto a(x^2)+x b(x^2).$$
A: I'll start my answer from a more general considerations. By $\mathbb{F}(S)$ I will denote a linear space over field $\mathbb{F}$ generated by basis $\{1_\mathbb{F}s:s\in S\}$.
In fact a linear isomorphism between $\mathbb{F}(S)$ and $\mathbb{F}(S)\oplus\mathbb{F}(S)$ exist iff ${\rm Card}(S)\geq\aleph_0$ or $S=\varnothing$
Indeed, if $S$ is finite, then ${\rm dim}(\mathbb{F}(S))={\rm Card}(\{1_\mathbb{F}s:s\in S\})={\rm Card(S)}$. Assume that there exist a linear isomorphism between $\mathbb{F}(S)$ and $\mathbb{F}(S)\oplus\mathbb{F}(S)$ then
$$
{\rm Card(S)}={\rm dim}(\mathbb{F}(S))={\rm dim}(\mathbb{F}(S)\oplus\mathbb{F}(S))={\rm dim}(\mathbb{F}(S))+{\rm dim}(\mathbb{F}(S))=2{\rm Card(S)}
$$
Hence ${\rm Card(S)}=0$ and $S$ must be an empty set. Indeed for $S=\varnothing$ we have $\mathbb{F}(S)=\{0\}$ and $\mathbb{F}(S)\oplus\mathbb{F}(S)=\{0\}\oplus\{0\}=\{0\}$
Now assume, that $S$ is infinite, i.e. ${\rm Card}(S)\geq\aleph_0$. Then there exist a lots of bijections between $S$ and $S\times S$. Consider one of this bijections $j:S\to S\times S$. Then consider standard inclusion 
$$
i:S\to\mathbb{F}(S):s\mapsto 1_{\mathbb{F}}s
$$
and standard projections 
$$
p_1:\mathbb{F}(S\times S)\to\mathbb{F}(S):\sum_k \lambda_k (s_{1,k},s_{2,k})\mapsto\sum_k \lambda_k s_{1,k}
$$
$$
p_2:\mathbb{F}(S\times S)\to\mathbb{F}(S):\sum_k \lambda_k (s_{1,k},s_{2,k})\mapsto\sum_k \lambda_k s_{2,k}
$$
Now the desired isomorphism (induced by $j$) is
$$
\phi:\mathbb{F}(S)\to\mathbb{F}(S)\oplus\mathbb{F}(S):\sum_k \lambda_k s_k\mapsto\sum_k \lambda_k p_1(i(j(s_k)))\oplus p_2(i(j(s_k)))
$$
Now we return to the original problem. In fact we can consider the linear space $\mathbb{F}[x]$ of polynomials of $x$ over field $\mathbb{F}$ as $\mathbb{F}(\mathbb{N})$. Indeed we have a linear isomorphism
$$
I:\mathbb{F}[x]\to\mathbb{F}(\mathbb{N}):\sum_k \lambda_k x^k\mapsto \sum_k \lambda_k k
$$
Since $\mathbb{F}[x]\sim \mathbb{F}(\mathbb{N})$ we can apply previous general result and obtain isomorphism between $\mathbb{F}[x]$ and $\mathbb{F}[x]\oplus\mathbb{F}[x]$
