Hi i'm trying to do an exercise of algebra and i'm stuck. Here it goes
Let $A$ be a factorial ring, $ F_n$ and $F_{n+1}$ two non-zero polynomials of $A[X_1,\cdots ,X_k](k\in \mathbb{N}^{*})$, homogenous of degree $n$ and $n+1$,respectively,$(n\in \mathbb{N})$ and $F=F_n+F_{n+1}$.
Show that, for any factorization $F=GH$, of the polynomials $G$ or $H$ is homogenous
Prove that $F$ is irreducible if $F_n$ and $F_{n+1}$ are relatively prime.
Show the reciprocal: if $F$ is irreducible then $F_n$ and $F_{n+1}$ are relatively prime.
For the first one i wanted to say :
Suppose we have a factorization $F=GH$ in $A[X_1, \cdots , X_k,T]$ then, \begin{align*} F(TX_1, \cdots , TX_k)&=F_n(TX_1, \cdots , TX_k)+F_{n+1}(TX_1, \cdots , TX_k) \\ &=T^nF_n(X_1, \cdots , X_k)+T^{n+1} F_{n+1}(X_1, \cdots , X_k) \text{ by homogeneity } \\ &=T^n((F_n(X_1, \cdots , X_k)+T^{1} F_{n+1}(X_1, \cdots , X_k)) \\ &=G(TX_1, \cdots , TX_k)H (TX_1, \cdots , TX_k) \end{align*} And for exemple we take $G(TX_1, \cdots , TX_k)=T^nG(X_1, \cdots , X_k)=T^n$ which is homogenous of degree n. Since $G$ is homogenous in $A[X_1, \cdots , X_k,T]$ from the following we know that $G $ will be homogenous of degree $n$ in $A[X_1, \cdots , X_k]$.
Is this correct ? For the others i would greatly appreciate any hints/solution/partial solutions to start ! I'm guessing we have to use the property of the ring $A$ but i can't figure out how.
Thank you for your time !