Solving advanced algebra related problems

This answer partially references GPT, GPT_Pro to solve the problem better
Proof: If f(x)g(x) is not all zero, then at least one of f(x) or g(x) is not zero.

Let f(x)=a_nx^n+a_(n-1)x^(n-1)+... +a_0,g(x)=b_mx^m+b_(m-1)x^(m-1)+... Plus b_0.

According to the formula:

(f(x)g(x)) ^n=(a_nx^n+a_(n-1)x^(n-1)+... +a_0)(b_mx^m+b_(m-1)x^(m-1)+... +b_0)^n
=a_n^nb_m^nx^{mn}+(a_nb_m^{n-1}+a_{n-1}b_m^n)x^{mn-1}+(a_nb_{m-1}^{n-1}+a_{n-1}b_{m-1}^{n-2}+... +a_0b_m^n)x^{mn-2}+... +(a_{n-1}b_0^{n-2}+a_0b_0^{n-1})x^{m+n-2}+a_0b_0^{n}x^{m+n-1}

and(f(x), g(x)) ^ n = a_nb_mx ^ {}(m + n) - 1 +(a_(n - 1} b_} {m - 1) x ^ {2} m + n - +(a_ b_ {n - 2} {2} m -) x ^ {3} m + n - +... +(a_0b_0)x^{m+n-(m+1)}

It follows that

When f(x) or g(x) is not zero, the coefficients of the above two formulas are the same, it can be proved.
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Lemma: Let $A$and $B$be elements of F[x], so ( A , B ) = ( g c d ( A , B ) ) , < / span > < / span > < / span > where $(A, B) $said A and B generated ideal.

Proof: Since $F[x]$is an Euclidean field, it is also a PID field, that is, every ideal can be generated by an element. Set $d = GCD(A, B) $, have the $d | A | B $$and $d, therefore $(A, B) \ subseteq(d) $. On the other hand, since $d$is the greatest common factor, it can be expressed as A linear combination of $A$and $B$, i.e. $d = aA + bB$, thus $d \in(a,B)$. Thus, $(d) \subseteq(A,B)$, i.e. $(A,B) =(d)$.

Now, let's go back to our equation. inherit; ( f ( x < /span> n , g ( x < /span> n < /span> = ( f ( x , g ( x ) < /span> n ) .

For any polynomial $a(x)$and $b(x)$, let $d = gcd(a(x),b(x))$, then according to the lemma, ( a ( Read the full article