Question

Let \(F\) be a field. Show that every non-zero ideal of \(F \llbracket X \rrbracket\) is of the form \(\left(X^{m}\right)\) for some uniquely determined integer \(m \geq 0\).

Short answer

Question: Prove that every non-zero ideal of the formal power series ring \(F \llbracket X \rrbracket\) is of the form \(\left(X^m\right)\) for some unique integer \(m \geq 0\). Answer: To prove this, we can follow these steps: 1. Show that any ideal of the form \(\left(X^m\right)\) is indeed a non-zero ideal of \(F \llbracket X \rrbracket\). This is true because the power series ring is closed under addition, multiplication and scalar multiplication, and every element of the ideal can be obtained by multiplying \(X^m\) by an element of \(F \llbracket X \rrbracket\). 2. For an arbitrary non-zero ideal \(I\) of \(F \llbracket X \rrbracket\), find the smallest power of \(X\) that belongs to \(I\), and call it \(X^m\). 3. Show that the ideal generated by \(X^m\), denoted as \(\left(X^m\right)\), is equal to \(I\). This can be done by proving that each element in \(I\) is a multiple of \(X^m\) and vice versa. 4. Prove that \(m\) is unique for each ideal. This can be done by showing that if \(\left(X^n\right) = I\), then \(m = n\). Therefore, every non-zero ideal of \(F \llbracket X \rrbracket\) is of the form \(\left(X^m\right)\) for some uniquely determined integer \(m \geq 0\).

Step-by-step solution

Show that \(\left(X^m\right)\) is a non-zero ideal of \(F \llbracket X \rrbracket\) for any \(m \geq 0\)

First, recall that an ideal consists of all the multiples of one or more elements. In this case, the elements are of the form \(X^m\), where \(m\) is a non-negative integer. Since the power series ring \(F \llbracket X \rrbracket\) is closed under addition, multiplication and scalar multiplication, every element of the ideal \(\left(X^m\right)\) can be obtained by multiplying \(X^m\) by an element of \(F \llbracket X \rrbracket\). As a result, \(\left(X^m\right)\) is an ideal for any \(m \geq 0\). Moreover, since \(X^m\) is a non-zero element, \(\left(X^m\right)\) is a non-zero ideal.

Find the smallest power of X in a non-zero ideal I

Let \(I\) be an arbitrary non-zero ideal of \(F \llbracket X \rrbracket\). Then there exists at least one non-zero element in \(I\) that is not a constant (since constant multiples form the ideal \(\left(1\right)\), which is the entire ring \(F \llbracket X \rrbracket\), and our goal is to show that such ideal is unique). Get the non-zero element in \(I\) with the smallest constant term and denote it as \(a_0 + a_1 X + a_2 X^2 + \cdots\), where \(a_0, a_1, a_2, \dots \in F\) and \(a_0 \neq 0\). Note that \(a_1, a_2, \dots\) can be zero or non-zero. Now, let \(m\) be the smallest non-negative integer such that \(a_m \neq 0\); that is, \(a_m\) is the first non-zero coefficient in the power series. Then \(X^m\) is the smallest power of \(X\) that belongs to \(I\).

Show that \(\left(X^m\right) = I\)

We need to show that \(\left(X^m\right)\) is equal to \(I\). To do this, we can prove that each element in \(I\) is a multiple of \(X^m\) and vice versa. Let \(b(X) = b_0 + b_1 X + b_2 X^2 + \cdots\) be an arbitrary element in \(I\). Since \(I\) is an ideal, we can write \(b(X) = (a_0 + a_1 X + a_2 X^2 + \cdots) f(X)\) for some \(f(X) \in F \llbracket X \rrbracket\). Observe that \(a_m\) divides \(b_m\) for all elements \(b(X) \in I\), and thus, \(b(X)\) must be a multiple of \(X^m\). Therefore, \(I \subseteq \left(X^m\right)\). On the other hand, consider an element \(c(X) = c_0 X^m + c_1 X^{m+1} + c_2 X^{m+2} + \cdots \in \left(X^m\right)\). We can rewrite it as \(c(X) = X^m (c_0 + c_1 X + c_2 X^2 + \cdots)\), with \(c_0 + c_1 X + c_2 X^2 + \cdots \in F \llbracket X \rrbracket\). Since \(I\) is an ideal and \(X^m\) is an element of \(I\), \(c(X)\) is also an element of \(I\). Therefore, \(\left(X^m\right) \subseteq I\). Since we have shown that \(I \subseteq \left(X^m\right)\) and \(\left(X^m\right) \subseteq I\), we conclude that \(\left(X^m\right) = I\).

Prove that m is unique for each ideal

Suppose there exists another integer \(n \geq 0\) such that \(\left(X^n\right) = I\). Taking an element from both ideals, we can write \(X^m = X^n f(X)\) for some \(f(X) \in F \llbracket X \rrbracket\). However, because both \(X^m\) and \(X^n\) are monomials (i.e., they have only one non-zero coefficient each), it must be the case that \(m = n\). This shows that \(m\) is unique for each ideal. Thus, every non-zero ideal of \(F \llbracket X \rrbracket\) is of the form \(\left(X^m\right)\) for some uniquely determined integer \(m \geq 0\).

Key concepts

Field Theory

Field theory is a branch of abstract algebra that deals with the study of fields. A field is a set equipped with two operations, addition and multiplication, that satisfies certain axioms similar to those of the familiar real numbers. Fields are integral to various areas of mathematics and its applications, including algebra, number theory, and algebraic geometry.

Central to field theory is the concept of a field extension, which is a larger field that contains a subfield. This concept plays an important role, especially when discussing the solvability of polynomial equations. One of the simplest examples of a field is the set of rational numbers, while more complex examples include the real numbers, complex numbers, and finite fields. Each field serves as a fertile ground for polynomial expressions and subsequently for the construction of rings such as the power series ring.

Additionally, field theory is pivotal in understanding the structure of ideals in ring theory, particularly in the context of power series rings. The characteristics of a field ensure that a ring of power series over a field exhibits particularly nice behaviors when it comes to its ideals, as explored in the given exercise.

Power Series Ring

The power series ring, denoted as \( F \llbracket X \rrbracket \), where \(F\) is a field, is an infinite series of the form \(a_0 + a_1X + a_2X^2 + \cdots\), with coefficients \(a_i\) in \(F\). Here, \(X\) often represents an indeterminate or a variable. The power series ring is an important object in mathematics as it extends the notion of polynomials and has applications in areas such as complex analysis, algebraic geometry, and formal algebra.

The power series ring shares many properties with the polynomial ring \(F[X]\), but it allows for an infinite number of non-zero terms. This accommodates concepts of convergence in the analytic sense when \(F\) is a complete field. Consequently, within the power series ring, we can perform operations like addition, multiplication, and composition, which are pivotal in formalizing various concepts in both pure and applied mathematics. It is also worth noting that power series rings over a field turn out to have a simple structure with respect to their ideals, which is beautifully demonstrated in the exercise related to non-zero ideals in such rings.

Non-zero Ideal

An ideal within ring theory is a subset of a ring that is closed under addition and multiplication by any element from the ring. It is a fundamental concept as it helps in defining the structure of rings, identifying their properties, and even categorizing them. When dealing with the power series ring, the ideals play a significant role in understanding how the elements in the ring relate to each other.

In the context of the exercise, a non-zero ideal is specifically one that contains at least one non-zero element. For the power series ring \( F \llbracket X \rrbracket \), every non-zero ideal has a special form: essentially, it is generated by a single element of the form \(X^m\), where \(m\) is a non-negative integer, and includes all multiples of that generating element. This simplification in the structure of ideals is unique to power series rings over a field and is exploited to solve equations and analyze polynomial expressions within the ring.

The non-triviality of non-zero ideals in the power series ring contrasts with the simplicity encountered in a field itself, where the only ideals are the zero ideal and the field itself. This distinctive characteristic of power series rings highlights the richness of structures that can arise even within the constraints of such specific algebraic objects.