Higher Algebra I — Fall 2005

Gerald Hoehn

Problem sheet 9 October 27, 2005

Problem 1: Let R be a commutative ring with 1. The ring of formal power series R[[x]] in the indeterminate x with coefficients from R, consists of formal infinite sums

∑_{n=0}^{∞} a_{i} x^{i} = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + · · · .

The ring structure is given by extending the ring structure of the polynomial ring R[x] to “polynomials of infinite degree” in the obvious way.

(a) Prove that ∑_{n=0}^{∞} a_{i} x^{i} is a unit in R[[x]] if and only if a_{0} is a unit in R.

(b) Prove that with R also R[[x]] is an integral domain.

Problem 2: Let R be a commutative ring with 1. Prove that a polynomial f ∈ R[x] is a zero-divisor in R[x] if and only if there exists a b ∈ R such that bf = 0.

Problem 3: Let D be a squarefree integer (i.e. an integer which has no integer square besides 1 as a divisor). Let

ω = \{ \begin{matrix}\sqrt{D}, & if D ≡ 2, 3 (mod 4), \\ ^{\frac{1+\sqrt{D}}{2}} , & if D ≡ 1 (mod 4). \end{matrix}

The set O = {a + b ω | a, b ∈ Z} is a subring of the real numbers (if D is positive) or complex numbers (if D is negative). Define the field norm of x + y\sqrt{D}, where x, y ∈ Q (or R), by N(x + y\sqrt{D}) = (x + y\sqrt{D})(x − y\sqrt{D}) = x^{2} − Dy^{2}.

Prove that for D = −2, −3, −7 or −11 the ring O is an Euclidean Domain with respect to N. (Hint: Modify the proof for Z[i] (D = −1) given in the lecture. For D = −3, −7, −11, prove that every element of C differs from an element of O by an element whose norm is at most (1 + |D|)^{2}/(16|D|), which is less than 1 for these values of D. Plotting the points of O in C may be helpful.)

Problem 4: Let K be a field, let a_{1}, …, a_{n} be pairwise different and let b_{1}, …, b_{n} be arbitrary elements of K. Show that there is a unique polynomial f ∈ K[x] with deg(f) ≤ n − 1 and f(a_{i}) = b_{i} for i = 1, …, n.