Question

Let \(k\) be a field. The ring of formal power series over \(k\), written \(k[[X]]\), is defined to be \(\left\\{\sum_{i=0}^{\infty} a_{i} X^{i} \mid a_{i} \in k\right\\}\). (As with polynomials, a rigorous definition is best given in terms of sequences \(\left(a_{0}, a_{1}, \ldots\right)\) of elements in \(k\); here we allow an infinite number of nonzero terms.) Define the sum by \(\sum a_{i} X^{i}+\sum b_{i} X^{i}=\Sigma\left(a_{i}+b_{i}\right) X^{i}\), and the product by \(\left(\sum a_{i} X^{i}\right)\left(\sum b_{i} X^{i}\right)=\sum c_{i} X^{i}\), where \(c_{i}=\sum_{j+k=i} a_{j} b_{k}\). Show that \(k[[X]]\) is a ring containing \(k[X]\) as a subring. Show that \(k[[X]]\) is a DVR with uniformizing parameter \(X\). Its quotient field is denoted \(k((X))\).

Short answer

Question: Prove that \(k[[X]]\) is a Discrete Valuation Ring (DVR) containing the polynomial ring \(k[X]\), and that its uniformizing parameter is \(X\). Answer: To prove that \(k[[X]]\) is a Discrete Valuation Ring (DVR) containing the polynomial ring \(k[X]\), we have followed the subsequent steps: Step 1: We defined addition and multiplication operations for power series. Step 2: We showed that with these operations, \(k[[X]]\) is a ring. Step 3: We demonstrated that \(k[X]\) is a subring of \(k[[X]]\). Step 4: We proved that \(k[[X]]\) is a DVR with uniformizing parameter \(X\) by showing that it is a local ring with a unique maximal ideal \((X)\) and satisfies the properties of a discrete valuation using the valuation function \(v\). Hence, \(k[[X]]\) is a Discrete Valuation Ring (DVR) containing the polynomial ring \(k[X]\) and has \(X\) as its uniformizing parameter.

Step-by-step solution

Define addition and multiplication operations

To add and multiply two power series, \(\sum a_i X^i\) and \(\sum b_i X^i\), we do the following: - Addition: Add the respective coefficients, i.e., \(\sum a_i X^i + \sum b_i X^i = \sum (a_i+b_i) X^i\) - Multiplication: Multiply the power series using the given rule, i.e., \(\left(\sum a_i X^i\right)\left(\sum b_i X^i\right) = \sum c_i X^i\), where \(c_i = \sum_{j+k=i} a_j b_k\)

Show that 𝑘[[𝑋]] is a ring

To prove that \(k[[X]]\) is a ring, we need to show that it satisfies the following properties: 1. Closure under addition and multiplication: From step 1, we can observe that the result of addition and multiplication remains in the form of the power series. Thus, \(k[[X]]\) is closed under addition and multiplication. 2. Associativity of addition and multiplication: As we defined addition and multiplication using the pointwise operations of coefficient elements in the field (in this case, \(k\)), both addition and multiplication operations are associative. 3. Commutativity of addition: Since the addition of coefficients in \(k\) is commutative, the addition of power series inherited the commutative property from the addition operation in fields. 4. Existence of additive identity: The zero power series, \(\sum 0 X^i\), serves as the additive identity. Adding it to any power series will not change the series. 5. Existence of additive inverses: For any power series \(\sum a_i X^i\), we can find its additive inverse by constructing another series \(\sum (-a_i) X^i\). The sum of these two series will be equal to the zero-power series \(\sum 0 X^i\). 6. Distributive property: Power series multiplication satisfies the distributive property since it is derived from the distributive property of field elements. By satisfying the above properties, \(k[[X]]\) is a ring.

Show that 𝑘[𝑋] is a subring of 𝑘[[𝑋]]

Polynomial ring \(k[X]\) is a collection of polynomials with coefficients in \(k\). Observe that a polynomial in \(k[X]\) can also be considered a power series with all coefficients \(a_i=0\) for \(i\geq n\) where \(n\) is the highest degree in the polynomial. Since the operations of addition and multiplication are consistent for both polynomial and power series, it is clear that \(k[X]\) is a subring of \(k[[X]]\).

Show that 𝑘[[𝑋]] is a DVR with uniformizing parameter 𝑋

To show that \(k[[X]]\) is a Discrete Valuation Ring (DVR) with uniformizing parameter \(X\), we need to demonstrate that it has the following properties: 1. Local ring: We need to show that \(k[[X]]\) contains a unique maximal ideal. Consider the ideal \((X)\) generated by \(X\). Every power series in \((X)\) has \(0\) as a constant term, which makes it a proper ideal. Moreover, all power series in \(k[[X]]\) but not in \((X)\) have invertible constant terms, so \((X)\) is a maximal ideal. 2. Discrete valuation: Define a valuation map \(v: k[[X]]\setminus{(X)} \longrightarrow \mathbb{Z}_{\geq 0}\) as follows: for a power series \(f(X)=\sum a_i X^i \in k[[X]]\) with \(a_0 \neq 0\), let \(v(f)\) be the minimum \(i\) for which \(a_i \neq 0\). This valuation function has the following properties: a. \(v(f)>0\) for any \(f\in (X)\), and \(v(f)=0\) for \(f \notin (X)\). b. \(v(f+g) \geq \min \{v(f), v(g)\}\) c. \(v(fg) = v(f)+v(g)\) By satisfying these properties, we can conclude that \(k[[X]]\) is a DVR with uniformizing parameter \(X\). As a final remark, the quotient field of our DVR, \(k[[X]]\), is the field of Laurent series \(k((X))\).

Key concepts

Algebraic Geometry and Formal Power Series

Algebraic geometry explores the properties of solutions to algebraic equations and their geometric manifestations. Formal power series fit into this framework as they provide a local perspective around a point — in our case, near the origin where the power of X is zero. Imagine zooming in very closely on a curve at a point to understand its immediate behavior; this is akin to examining the series expansion at that point. In formal power series rings like k[[X]], we can study properties of algebraic objects such as singularities, which are crucial for understanding the broader geometric structure. This local examination is foundational in algebraic geometry, as it allows for the extrapolation of global properties from local observations.

Understanding the operations in k[[X]] — addition and multiplication of series — is key to exploring the geometry of algebraic curves. Each term in a power series expansion can be thought of as contributing to the 'shape' of the curve at an infinitesimally close range to a point. These series are thus invaluable tools for algebraic geometers as they can capture subtle geometric nuances that polynomials alone might miss.

Discrete Valuation Ring (DVR)

Understanding DVR through k[[X]]

A Discrete Valuation Ring (DVR) is a type of local ring important in number theory and algebraic geometry, known for its valuation property, which measures 'how divisible' an element is by a fixed generator of the DVR’s maximal ideal. In our exercise, k[[X]] is shown to be a DVR. The ring’s maximal ideal is generated by X, and the valuation is a way to count the powers of X that divide a given series. This is akin to how you might measure the divisibility of integers by prime numbers, but instead, we are measuring the divisibility of series terms by X.

Local rings like DVRs have a single 'path' from any point to their 'center,' reflecting how a DVR has a single maximal ideal. This quality makes them particularly useful in singularity analysis in algebraic geometry and for defining and working with local properties of algebraic structures.

Polynomial Ring k[X]

The polynomial ring k[X] is one of the simplest examples of a ring that is not a field. It contains all polynomials with coefficients in the field k. When considering polynomial rings, it's meaningful to look at them in the context of formal power series. By seeing polynomials as finite power series where all but a finite number of coefficients are zero, we bridge the gap between understanding discrete algebraic objects and more fluid, possibly infinite, algebraic structures as seen in formal power series.

This perspective enriches our understanding and provides a more holistic view of the algebraic landscape, knowing that the same rules that apply for finite combinations in the polynomial ring extend to infinite combinations in the formal power series ring.

Quotient Field of DVR

The Leap from Ring to Field

The quotient field of a Discrete Valuation Ring (DVR) like k[[X]] is the set of all possible fractions where the numerator and denominator are elements of the DVR, excluding the zero denominator. For our specific case, k[[X]] leads to the field of Laurent series, denoted by k((X)). Laurent series can be thought of as 'two-sided' power series including terms with negative exponents.

This extension from DVR to its quotient field mirrors the step from integers to rational numbers. Just as every non-zero integer has a multiplicative inverse in the rationals, every non-zero element in a DVR has an inverse in its quotient field. The field k((X)) immensely expands our playground, allowing algebraists and geometers to manipulate power series with even greater flexibility, which is essential for many theoretical and practical applications in algebra and beyond.