Exercise 4.2

Question

Suppose $m$ is a positive integer. Is the set
$$
\{0\} \cup \{p \in \mathcal{P}(F): 	ext{deg} p = m\}
$$
a subspace of $\mathcal{P}(F)$?

Arden Shi · Accepted Answer

Proof: 
Let the space in the problem be known as $U$.
Then, we want to prove closure upon addition and multiplication. Then, $\forany p, q \in U 	ext{ and } \uplambda \in \mathds{C}$, we want to show $p+q \in U$ and $\uplambda p \in U$.
First, define $a_0, ..., a_m$ and $b_0, ..., b_m \in \mathds{C}$ where
$$
p = a_m x^m + \dotsb + a_0
$$
and,
$$
p = b_m x^m + \dotsb + b_0.
$$
Then to prove additive closure, 
$$
p+q = (a_m + b_m) x^m + \dotsb + (a_0 + b_0)
$$
which clearly also has degree of either $m$ or $0$.
For multiplicative closure,
$$
\uplambda p = (a_m \uplambda) x^m + \dotsb + (a_0 \uplambda)
$$
which also clearly has degree $m$ or $0$.
Hence, due to its closure additively and multiplicatively, $U$ is a subspace.

Exercise 4.2

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Exercise 4.2

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