1. Let \(K\) be a field and let \(R=K[x]\) be the set of all polynomials in one variable over \(K\). So an element of \(R\) is a finite formal sum \(\sum_{i=0}^{d} a_{i} x^{i}\) with \(a_{i} ∈ K\).
   If \(f=\sum_{i} a_{i} x^{i}\) and \(g=\sum_{j} b_{j} x^{j}\) then define \(f+g\) and \(f × g\) in the obvious way: \(f+g=\) \(\sum_{i}\left(a_{i}+b_{i}\right) x^{i}\) and \(f g=\sum_{k} c_{k} x^{k}\) with \(c_{k}=\sum_{i+j=k} a_{i} b_{j}\).
   Prove that \(R\) becomes a commutative ring with a 1 with these definitions of + and \(×\).
2. • Prove that if \(R\) is a commutative ring with a 1 , and \(x ∈ R\) then \(0 x=0\).
   • Prove that if \(K\) is a field and \(a, b ∈ K\) are both non-zero, then \(a b ≠ 0\).
   • If \(K\) is a field and \(f=\sum_{i=0}^{d} a_{i} x^{i} ∈ K[x]\) is a non-zero polynomial, then (by choosing \(d\) sensibly) we may assume \(a_{d} ≠ 0\); we call \(a_{d} x^{d}\) the leading term of \(f\), and \(d\) the degree of \(f\), and we write \(d=\deg(f)\). Prove that if \(f, g ∈ K[x]\) are non-zero, then \(f g\) is also non-zero, and \(\deg(f g)=\deg(f)+\deg(g)\).
   • Prove that if \(f, g, h ∈ K[x]\) and \(h ≠ 0\) and \(f h=g h\), then \(f=g\) (the cancellation property for polynomial rings).
     [Those doing Algebra III will soon learn that \(f, g ≠ 0 ⟹ f g ≠ 0\) is the assertion that \(K[x]\) is an integral domain.]
3. Let's goof around in \(ℚ[x]\).
   • Find the quotient and remainder when \(x^{5}+x+1\) is divided by \(x^{2}+1\).
   • Find the remainder when \(x^{1000}+32 x^{53}+8\) is divided by \(x-1\) (hint: use your head instead of just calculating).
   • Find polynomials \(s(x)\) and \(t(x)\) such that \[ \left(2 x^{3}+2 x^{2}+3 x+2\right) s(x)+\left(x^{2}+1\right) t(x)=1 . \] [bonus question: I just made those two polynomials above up. How did I know for sure that they were coprime?]
   • Find a gcd for \(x^{4}+4\) and \(x^{3}-2 x+4\). Express it as \(a(x)\left(x^{4}+4\right)+b(x)\left(x^{3}-2 x+4\right)\).
   • Find polynomials \(λ(x)\) and \(μ(x)\) such that \((1+x) λ(x)+\left(x^{3}-2\right) μ(x)=1\).
   • Find rational numbers \(a, b\) and \(c\) such that if \(α=\sqrt[3]{2}\) then \(a+b α+c α^{2}=1 /(1+α)\). Hint: use (e).
4. Prove that if \(f, g ∈ K[x]\) and at least one is non-zero, and if \(s, t\) are both gcd's of \(f\) and \(g\), then \(s=λ t\) for some \(λ ∈ K^{×}\).
5. • We know that whether or not a polynomial is irreducible depends on which field it's considered as being over – for example \(x^{2}+1\) is irreducible in \(ℚ[x]\) but not in \(ℂ[x]\). But show that the notion of divisibility does not depend on such issues. More precisely show that if \(K ⊆ L\) are fields, if \(f, g ∈ K[x]\), and if \(f ∣ g\) in \(L[x]\) then \(f ∣ g\) in \(K[x]\).
   • Is it true that if \(f, g ∈ ℤ[x]\) and \(f ∣ g\) in \(ℚ[x]\) then \(f ∣ g\) in \(ℤ[x]\) ? [hint: no]. Is it true under the extra assumption that \(f\) is monic? [hint: yes]
6. Factor the following polynomials in \(ℚ[x]\) into irreducible ones, giving proofs that your factors really are irreducible.
   • \(x^{3}-8\)
   • \(x^{1000}-6\)
   • \(x^{4}+4\) (hint: Q3)
   • \(2 x^{3}+5 x^{2}+5 x+3\)
   • \(x^{5}+6 x^{2}-9 x+12\)
   • \(x^{73}-1\)
   • \(x^{73}+1\)
   • \(x^{12}-1\) [NB this one seems quite tricky.]