I'm studying Hungerford's introductory book on algebra, section 4.4. I'm a bit confused about what the question is specifically asking me to do
- Let $f(x),g(x),h(x) \in F[x]$ and $a\in F$.
(a) If $f(x)=g(x)+h(x)$ in $F[x]$, show that $f(a)=g(a)+h(a)$ in $F$.
(b) If $f(x)=g(x)h(x)$ in $F[x]$, show that $f(a)=g(a)h(a)$ in $F$.
This is how I'm thinking about it (although I know it isn't quite right):
Since $f(x)$ and $g(x)+h(x)$ are the same polynomials in $F[x]$, they must induce the same function from $F$ to $F$, hence $f(a)=g(a)+h(a)$.
Another related problem that I'm having difficulty with is 27:
- Let $T$ be the set of all polynomial functions from $F$ to $F$. Show that $T$ is a commutative ring with identity, with operations defined as in calculus: For each $r \in F$, $$(f+g)(r)=f(r)+g(r)$$ and $$(fg)(r)=f(r)g(r)$$
To show that $T$ is closed under addition, let $f,g \in T$. Since $f,g:F \to F$, for any $r \in F$ we have $f(r),g(r) \in F$. Since $F$ is a field, $F$ is closed under addition. Hence, $f(r)+g(r) \in F$ for every $r \in F$.
The problem also hints: use Exercise 23 to verify that $f+g$ and $fg$ are the polynomial functions induced by the sum and product polynomials $f(x)+g(x)$ and $f(x)g(x)$, respectively.
But I am not quite sure exactly where this fits in the proof that T is a ring.
I understand that the book is trying to draw a distinction between polynomials (as elements of a ring $R[x]$, or $F[x]$) and polynomial functions based on the distinction between $x$ as an indeterminate and $x$ as a variable, but I'm a little confused about going between the two different uses. Even though I think I am understanding the main ideas so far in this book, I think there are more subtle points in the text that I'm a bit confused on or oblivious of and this section is making me doubt how well I really comprehend it.
This is what I have so far:
(a) Suppose $g(x)= \sum_{i=0}^n p_i x^i$, where $n$ is a nonnegative integer and $p_i \in \mathit F $ for each $i$, and likewise $h(x)= \sum_{j=0}^m q_j x^j$, where $m$ is a nonnegative integer and $q_j \in \mathit F$ for each $j$.
Then $f(x)=g(x)+h(x)= \sum_{k=0}^l (p_k+q_k)x^k$, where $l=max(m,n)$, by the definition of polynomial addition. Therefore, evaluating $f(x)$ at $a$ and using the commutative, associative, and distributive properties of $ \mathit F$: $$f(a)= \sum_{k=0}^l (p_k+q_k)x^k=( \sum_{k=0}^l p_k a^k)+( \sum_{k=0}^l q_k a^k)=( \sum_{i=0}^n p_i a^i )+( \sum_{j=0}^m q_j a^j)=g(a)+h(a).$$ (b) By the definition of polynomial multiplication, $f(x)=g(x)h(x)= \sum_{k=0}^{m+n} ( \sum_{i+j=k} p_i q_j)x^k$. Evaluating $f(x)$ at $a$ and then using the commutative, associative, and distributive properties of $ \mathit F$:
$$f(a)= \sum_{k=0}^{m+n} ( \sum_{i+j=k} p_i q_j)a^k=( \sum_{i=0}^n p_i a^i)( \sum_{j=0}^m q_j a^j)=g(a)h(a).$$ 24. Problem 23 shows that if $f(x)=g(x)+h(x)$ or $f(x)=g(x)h(x)$ in $ \mathit F[x]$, then $f(a)=g(a)+h(a)$ or $f(a)=g(a)h(a)$ in $ \mathit F$ respectively. Therefore, $$φ_a(g(x)+h(x))=g(a)+h(a)=φ_a(g(x))+φ_a(h(x))$$ and $$φ_a(g(x)h(x))=g(a)h(a)=φ_a(g(x))φ_a(h(x))$$ Hence, $φ_a$ is a homomorphism.