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Magazine Problem 4
Prove that deg \(a(x, y) b(x, y)=\operatorname{deg} a(x, y)+\operatorname{deg} b(x, y)\) if \(A\) is an integral domain.
Problem 4
Show that the following is true in \(A[x]\) for any ring \(A:\) For any odd \(n\), (a) \(x+1\) is a factor of \(x^{n}+1\) (b) \(x+1\) is a factor of \(x^{n}+x^{n-1}+\cdots+x+1\)
Problem 4
Let \(g: A[x] \rightarrow A\) send every polynomial to the sum of its coefficients. Prove that \(g\) is a surjective homomorphism, and describe its kernel.
Problem 5
If \(A\) is an integral domain, we have seen that in \(A[x]\), $$ \operatorname{deg} a(x) b(x)=\operatorname{deg} a(x)+\operatorname{deg} b(x) $$ Show that if \(A\) is not an integral domain, we can always find polynomials \(a(x)\) and \(b(x)\) such that \(\operatorname{deg} a(x) b(x)<\operatorname{deg} a(x)+\operatorname{deg} b(x)\)
Problem 5
Prove the following: In \(\mathbb{Z}_{3}[x], x+2\) is a factor of \(x^{m}+2\), for all \(m .\) In \(\mathbb{Z}_{n}[x]\), \(x+(n-1)\) is a factor of \(x^{m}+(n-1)\), for all \(m\) and \(n\).
Problem 5
Find an example of each of the following in \(\mathbb{Z}_{8}[x]:\) a divisor of zero, an invertible element, an idempotent element.
Problem 6
Prove that there is no integer \(m\) such that \(3 x^{2}+4 x+m\) is a factor of \(6 x^{4}+50\) in \(\mathbb{Z}[x]\)
Problem 6
Explain why \(x\) cannot be invertible in any \(A[x]\), hence no domain of polynomials can ever be a field.
Problem 6
Show that if \(A\) is an integral domain, the only invertible elements in \(A[x]\) are the constant polynomials \(\pm 1\). Then show that in \(\mathbb{Z}_{4}[x]\) there are invertible polynomials of all degrees.
Problem 6
If \(h: \mathbb{Z} \rightarrow \mathbb{Z}_{n}\) is the natural homomorphism, let \(\bar{h}: \mathbb{Z}[x] \rightarrow \mathbb{Z}_{n}[x]\) be the homomorphism induced by \(h\). Prove that \(\bar{h}(a(x))=0\) iff \(n\) divides every coefficient of \(a(x)\).
