Question

Let \(R\) be an integral domain with field of fractions \(K\) and \(a, b \in R|x|\) of degree \(n \geq m \geq 0\). Then we can apply the polynomial division algorithm \(2.5\) to compute \(q, r \in K|x|\) such that \(a=q b+r\) and deg \(r<\) deg \(b\). (i) Prove that there exist \(q, r \in R[x]\) with \(a=q b+r\) and \(\operatorname{deg} r<\mathrm{deg} b\) if and only if \(\mathrm{lc}(b) \mid \mathrm{lc}(r)\) in \(R\) every time the algorithm passes through step 3, and that they are unique in that case. (ii) Modify Algorithm \(2.5\) so that on input \(a, b\), it decides whether \(q, r \in \boldsymbol{R}|x|\) as in (i) exist, and if so, computes them. Show that this takes the same number of operations in \(R\) as given in the text, where one operation is cither an addition or a multiplication in \(R\), or a test which decides whether an element \(c \in R\) divides another element \(d \in R\), and if so, computes the quotient \(d / c \in R\).

Short answer

Existence and uniqueness occur if \( \mathrm{lc}(b) \mid \mathrm{lc}(r) \) throughout; the modified algorithm checks this divisibility.

Step-by-step solution

Understand the Polynomial Division Algorithm

First, we need to understand the concept of dividing a polynomial \(a\) by another polynomial \(b\), which results in a quotient \(q\) and a remainder \(r\) such that \(a = qb + r\) and \(\deg(r) < \deg(b)\). This is similar to long division for numbers.

Examine Leading Coefficient Conditions

To determine if the polynomial division results in quotients and remainders within \(R[x]\), observe that at each division step, the leading coefficient of the remainder \(\mathrm{lc}(r)\) must be divisible by the leading coefficient of \(b\), \(\mathrm{lc}(b)\), in \(R\). This ensures that \(q\) and \(r\) end up being in \(R[x]\).

Prove Existence and Uniqueness of \(q\) and \(r\)

Assuming \(\mathrm{lc}(b)\) divides \(\mathrm{lc}(r)\) at every step, it means at each reduction, the elements in the polynomials remain in \(R\). This leads to the unique construction of \(q\) and \(r\), as each step incrementally constructs the coefficients of \(q\) and reduces the degree of \(r\) as needed.

Modify Algorithm 2.5 for \(R[x]\)

Transform the algorithm so that when dividing terms, ensure each division checks if \(\mathrm{lc}(b)\) divides the necessary coefficients, modifying the standard polynomial division to handle this condition. This modification involves checking divisibility while constructing \(q\) and \(r\), ensuring they remain in \(R[x]\).

Consider Complexity and Operations

The modified algorithm retains operations similar to the basic polynomial division, as it primarily involves additional divisibility checks which can be computed in-place. Therefore, the complexity stays similar, focusing on multiplications, additions, and the divisibility test (and division) where needed.

Key concepts

Integral Domain

An integral domain is an important concept in algebra. It is a special type of commutative ring but with no zero divisors. This means that if you multiply two non-zero elements, you cannot get zero. This property is crucial when dealing with polynomial division, as it ensures that you can safely divide polynomials without running into problems with undefined divisions.

• Commutative ring: A set with addition and multiplication, similar to the integers but generalized.
• No zero divisors: A property ensuring multiplication behaves almost like with numbers, no unexpected zeros.
• Division safe: Because of the no zero divisor rule, dividing remains consistent and predictable.

This quality of integral domains makes them very suitable for algebraic operations like polynomial division, ensuring calculations stay within defined limits.

Field of Fractions

The field of fractions is an extension concept in algebra. For an integral domain, the field of fractions is the smallest field in which the domain can be embedded. Essentially, it means taking your original integral domain and adding all possible "fractions" to it.

• Field: A structure where you can add, subtract, multiply, and divide (except by zero).
• Fractions: Like ordinary fractions, a ratio of two elements from your domain.
• Embedding: A way to fit your smaller structure into a larger one without losing the crucial properties.

This field helps in polynomial division as sometimes you need to consider fractions of coefficients, which the field of fractions easily accommodates, making the mathematics more flexible and powerful.

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. In polynomial division, particularly in integral domains, the leading coefficient plays a pivotal role. It helps determine the nature of division and guides the solution process.

• Highest degree term: This is the term in the polynomial with the highest power of the variable.
• Crucial for division: Determines how to scale down polynomials during division.
• Guides uniqueness: In the modified algorithm, ensuring leading coefficient divisibility is key.

By focusing on the leading coefficients, one ensures that each division step results in valid, manageable polynomials, which are a significant part of the division feasibility and correctness in the integral domain.

Existence and Uniqueness

Existence and uniqueness are fundamental properties in mathematics ensuring a solution can be found and is distinct. In the context of polynomial division in an integral domain, ensuring these properties is crucial.

• Existence: The solution (in this case, polynomials \(q\) and \(r\)) exists under given conditions.
• Uniqueness: The solution produced is the only one possible under the rules set.
• Condition: The divisibility of leading coefficients at each step ensures existence and uniqueness.

Establishing existence and uniqueness requires verifying that these properties hold throughout the polynomial division process, and is central to modifying algorithms to meet these conditions.

Algorithm Modification

Algorithm modification involves adjusting an existing computational process to achieve new results or efficiencies. For polynomial division in integral domains, this involves ensuring any proposed algorithm respects the properties of the domain.

• Ensures proper checks: Divisibility checks for leading coefficients are a new algorithm addition.
• Maintains efficiency: While adding checks, it preserves the operational efficiency, meaning complexity remains manageable.
• Proof of correctness: Any modification should come with a proof that it actually works in the intended way.

These modifications ensure that polynomial division remains consistent and reliable across a broader set of conditions, making solutions robust and applicable to more complex algebraic problems.