Problem 1 - A Book of Abstract Algebra

Question

\(A\left[x_{1}, x_{2}\right]\) denotes the ring of all the polynomials in two letters \(x_{1}\) and \(x_{2}\) with coefficients in \(A\). For example, \(x^{2}-2 x y+y^{2}+x-5\) is a quadratic polynomial in \(\mathbb{Q}[x, y] .\) More generally, \(A\left[x_{1}, \ldots, x_{n}\right]\) is the ring of all the polynomials in \(n\) letters \(x_{1}, \ldots, x_{n}\) with coefficients in \(A .\) Formally it is defined as follows: Let \(A\left[x_{1}\right]\) be denoted by \(A_{1} ;\) then \(A_{1}\left[x_{2}\right]\) is \(A\left[x_{1}, x_{2}\right]\). Continuing in this fashion, we may adjoin one new letter \(x_{i}\) at a time, to get \(A\left[x_{1}, \ldots, x_{n}\right]\) 1 Prove that if \(A\) is an integral domain, then \(A\left[x_{1}, \ldots, x_{n}\right]\) is an integral domain.

Step-by-Step Solution

Answer

By induction, if \( A \) is an integral domain, then \( A[x_1, ..., x_n] \) is also an integral domain.

Step 1: Understand the Definition of Integral Domain

An integral domain is a commutative ring with unity (1≠0) and no zero divisors. This means that in an integral domain, multiplying two non-zero elements always gives a non-zero product.

Step 2: Base Case: Polynomial in One Variable

Start with the base case. If \( A \) is an integral domain, then \( A[x_1] \) is also an integral domain. This is because the polynomial ring in one variable with coefficients from an integral domain is known to retain the properties of being an integral domain.

Step 3: Inductive Step Hypothesis

Assume that \( A[x_1, x_2, ..., x_{k-1}] \) is an integral domain for some integer \( k \geq 1 \). This means that there are no zero divisors when considering polynomials in the variables \( x_1, x_2, ..., x_{k-1} \).

Step 4: Inductive Step: Adjoin one More Variable

We need to show that \( A[x_1, x_2, ..., x_{k}] \) is also an integral domain. Consider two non-zero elements \( f(x_1, ..., x_k) \) and \( g(x_1, ..., x_k) \) in \( A[x_1, x_2, ..., x_{k}] \). They are polynomials in \( x_k \) with coefficients in \( A[x_1, x_2, ..., x_{k-1}] \). The leading coefficient of a product \( f(x_k)g(x_k) \) is the product of the leading coefficients of \( f \) and \( g \). Both leading coefficients are non-zero (due to the hypothesis), thereby the product is non-zero.

Step 5: Conclusion of the Inductive Proof

Since there are no zero divisors in \( A[x_1, x_2, ..., x_{k-1}] \) and we have shown that multiplying two non-zero elements in \( A[x_1, x_2, ..., x_{k}] \) also results in a non-zero element, \( A[x_1, x_2, ..., x_{k}] \) is an integral domain.

Key Concepts

Polynomial Ring

Polynomial rings are a fascinating concept in algebra. They are constructed by taking coefficients from a commutative ring and forming expressions called polynomials. These expressions have variables, like \( x_1, x_2, \ldots, x_n \), and the coefficients come from a base ring \( A \). In simple terms, imagine taking numbers from a set (like the integers) and using them to create expressions with variables, such as \( 2x^2 + 3x + 5 \). Here, '2', '3', and '5' are coefficients from the set of integers.

You can build these polynomial rings step by step. Start by considering expressions in just one variable, called \( A[x_1] \). Here, \( x_1 \) acts as a placeholder that doesn't have a fixed value. Then you can move to polynomials in two variables, \( A[x_1, x_2] \), and so on. This process can be extended to polynomials in \( n\) variables, \( A[x_1, \ldots, x_n] \). The variables allow us to create more complex expressions, and exploring their properties can reveal deep insights about algebraic structures.

Commutative Ring

At its core, a commutative ring is a set equipped with two operations: addition and multiplication. These operations satisfy several important properties. First, the operations themselves are both commutative, meaning that for any two elements \( a \) and \( b \), \( a + b = b + a \) and \( a \times b = b \times a \). Second, there is an additive identity, which usually is the number '0'. Lastly, for multiplication, there is an identity element like '1' in integers, meaning any element multiplied by this identity gives back the same element.

Besides these fundamental properties, all elements in a commutative ring have additive inverses. This means that for any element \( a \), there is an element \( -a \) such that \( a + (-a) = 0 \). Understanding commutative rings is crucial when diving into polynomial rings and other algebraic structures because it lays down the ground rules for how elements can interact.

Inductive Proof

Inductive proof is a powerful method used in mathematics to prove statements that are asserted for all natural numbers. It starts with the base case, which is the simplest instance. Once the base case is shown to be true, you proceed to the inductive step.

For the inductive step, assume the statement holds for a particular case, say \( n = k \). This assumption is called the inductive hypothesis. Then, show that the statement holds for the next case, \( n = k + 1 \). By establishing this, the truth of the statement is "inducted" up, meaning it is true for all numbers. This method is especially helpful in proving properties that are built layer by layer, as in the case of polynomial rings with one variable extending to two, then three, and so on.

Zero Divisors

Zero divisors are one of those hidden snags in algebra that can disrupt many of the properties we take for granted. In a ring, an element \( a \) is a zero divisor if there's a non-zero element \( b \) such that \( a \times b = 0 \). This is important because zero divisors break the integral domain's rule that the product of any two non-zero elements cannot be zero.

Integral domains, which are special types of commutative rings, specifically forbid zero divisors. This property is key in keeping arithmetic simple and predictable, much like in the ordinary numbers we first learn about. By ensuring a ring is free of zero divisors, we preserve the nice feature that non-zero elements won't unexpectedly create zero when multiplied, facilitating solid reasoning especially in polynomial rings.