Let \(\QQ\) be the field of rational numbers. Determine which of the following subsets of \(Q[x]\) are ideals. When the set is an ideal, find its monic generator.
all \(f\) of even degree;
all \(f\) of degree \(\geq 5\);
all \(f\) such that \(f(0) = 0\);
all \(f\) such that \(f(2) = f(4) = 0\);
all \(f\) in the range of the linear operator \(T\) defined by \[ T\left(\sum_{i=0}^{n} c_i x^i\right) = \sum_{i=0}^{n} \frac{c_i}{i+1} x^{i+1}. \]
Solution
(a) Not an ideal.
1. \(x^2\) has even degree, but \(x^2 \cdot x = x^3\) has odd degree.
the set of even-degree polynomials is not closed under multiplication by \(x\).
2. Hence the set of even-degree polynomials is not an ideal.
1 (an ideal is closed under multiplication by any ring element).
(b) Not an ideal.
1. \(x^5\) has degree \(5 \ge 5\), but \(x^5 \cdot x^{-1}\) is not a polynomial, and more relevantly \(x^5 - x^5 = 0\) has degree \(-\infty < 5\).
the set is not closed under subtraction (the zero polynomial is not in it).
2. Hence the set of polynomials of degree \(\ge 5\) is not an ideal.
1 (an ideal must contain \(0\) and be closed under subtraction).
(c) An ideal, generated by \(x\).
1. The set \(\{f : f(0) = 0\}\) is the kernel of the evaluation map \(\operatorname{ev}_0 : \QQ[x] \to \QQ\), \(f \mapsto f(0)\).
definition.
2. Hence it is an ideal.
the kernel of a ring homomorphism is an ideal.
3. \(f(0) = 0\) iff \(x \mid f\), so the ideal is \((x)\).
a polynomial vanishes at \(0\) iff it has no constant term iff it is divisible by \(x\).
4. The monic generator is \(x\).
3.
(d) An ideal, generated by \((x-2)(x-4)\).
1. The set \(\{f : f(2) = f(4) = 0\}\) is the kernel of the map \(\QQ[x] \to \QQ \times \QQ\), \(f \mapsto (f(2), f(4))\).
definition.
2. Hence it is an ideal.
kernel of a ring homomorphism.
3. \(f(2) = f(4) = 0\) iff \((x-2)(x-4) \mid f\).
\(f(2) = 0\) iff \((x-2) \mid f\), and \(f(4) = 0\) iff \((x-4) \mid f\); since \((x-2)\) and \((x-4)\) are coprime, both hold iff their product divides \(f\).
4. The monic generator is \((x-2)(x-4)\).
3.
(e) An ideal, generated by \(x\).
1. \(T\) is the linear operator sending \(x^i \mapsto \frac{1}{i+1} x^{i+1}\).
definition.
2. The range of \(T\) is \(\{f \in \QQ[x] : f(0) = 0\}\), i.e. the polynomials with no constant term.
\(T\) raises degree by \(1\), so every output is divisible by \(x\); conversely every \(x^{i+1}\) (for \(i \ge 0\)) is in the range, so the range is exactly the ideal \((x)\).
3. Hence the range is an ideal, with monic generator \(x\).
2 and (c).
4. Q.E.D.
2 (a), 2 (b), 4 (c), 4 (d), 3 (e).