Exercise 1.3.25

Question

Let $\mathrm{W}_{1}$ denote the set of all polynomials $f(x)$ in $\mathrm{P}(F)$ such that in the representation
$$
f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0},
$$
we have $a_{i}=0$ whenever $i$ is even. Likewise let $\mathrm{W}_{2}$ denote the set of all polynomials $g(x)$ in $\mathrm{P}(F)$ such that in the representation
$$
g(x)=b_{m} x^{m}+b_{m-1} x^{m-1}+\cdots+b_{1} x+b_{0},
$$
we have $b_{i}=0$ whenever $i$ is odd. Prove that $\mathrm{P}(F)=\mathrm{W}_{1} \oplus \mathrm{W}_{2}$.

Jephian C.-H. Lin · Accepted Answer

This is similar to \href{./exercise_1-3-24}{Exercise 1.3.24}.

Exercise 1.3.25

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Exercise 1.3.25

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