Question

For each \(n \in \mathbb{N}\), find polynomials \(f_{1} \ldots, f_{n} \in Q\left[x_{1}\right.\) such that \(\operatorname{ged}\left(f_{1} \ldots \ldots f_{n}\right)=1\) and any proper subset of them has a nonconstant ged.

Short answer

Use distinct linear polynomials \(f_i(x_1) = x_1 + c_i\) for each \(n\).

Step-by-step solution

Understand the Exercise

The problem requires finding a set of polynomials \(f_1, f_2, \ldots, f_n\) such that the greatest common divisor of all \(f_i\) is 1, but every proper subset has a greater gcd that is not a constant.

Basic Case Exploration

Start with the simplest case, \(n=2\). Consider the polynomials \(f_1(x_1) = x_1\) and \(f_2(x_1) = x_1 + 1\). The gcd of two polynomials \(x_1\) and \(x_1 + 1\) is 1. If any subset, like the entire set, is considered, each polynomial individually is nonconstant.

Examine the Condition for Larger n

We build this idea for larger \(n\). Assume for \(n\), each \(f_i(x_1) = x_1 + c_i\) where \(c_i\) are distinct integers, ensuring that \( \gcd(f_1(x_1), f_2(x_1), \ldots, f_n(x_1)) = 1\). Proper subsets will include at least two such distinct polynomials, with gcd other than 1.

Construct Specific Example

For a specific \(n\), construct polynomials \(f_1(x_1) = x_1, f_2(x_1) = x_1 + 1, \ldots, f_n(x_1) = x_1 + n-1\). Here, any pair of polynomials share no common non-constant divisor, keeping subsets nonconstant.

Verify Conditions

For the set \(\{f_1(x_1), f_2(x_1), \ldots, f_n(x_1)\}\), ensure the full set has \( \gcd = 1 \), and any subset of at least two elements with same form (distinct linear factors) still maintains a gcd greater than 1 if at least two are chosen with distinct offsets.

Key concepts

Greatest common divisor

The greatest common divisor (GCD) is a fundamental concept in mathematics, especially when dealing with numbers and polynomials. It refers to the largest polynomial, or in the case of numbers, the largest integer, that divides two or more polynomials without leaving a remainder. In the context of polynomials, this means finding the polynomial of the highest degree that is a factor of all given polynomials.

When looking for the GCD of multiple polynomials, we generally follow these steps:

• Identify and factor each polynomial.
• Find the common factors across all polynomials.
• Select the highest degree polynomial that appears in all factor lists.

In our specific exercise, we are asked to find a set of polynomials such that the GCD of the entire set is 1. This indicates that while the polynomials in the set are related, they do not have any non-trivial polynomial factors in common. However, any proper subset (not including the whole set) should have a nonconstant GCD, meaning they share some polynomial factors not equal to just 1.

Polynomial subsets

When dealing with polynomial subsets, we focus on selecting portions of a larger set to analyze their properties. In the original exercise, subsets refer to any group of polynomials that can be drawn from the entire given set. Understanding the properties of these subsets helps in exploring the relationships and interactions between the polynomials.

For our problem, the interest is in ensuring that any subset of the chosen polynomials—meaning any selection that doesn't include every polynomial in the original list—has a nonconstant GCD. This indicates that while no single polynomial in the entire set is fully divisible by any other, there's some overlap or shared structure in smaller groups.

By using specific forms like linear polynomials with distinct complements (e.g., derived terms like \(x_1 + c_i\) where each \(c_i\) is distinct), we ensure that small subset combinations meet the criteria of having a meaningful GCD. This strategy ensures that while the entire group of polynomials looks fairly unrelated, except for sharing the variable structure, smaller groups reveal a deeper connection through their divisors.

Distinct linear polynomials

Linear polynomials are a simpler form of polynomial, usually represented by expressions like \(ax + b\). In our exercise, we focus on creating distinct linear polynomials. "Distinct" in this context means that each polynomial differs in its constant term or coefficient.

The significance of using distinct linear polynomials is that with each different constant term, the greatest existing overlap (or common divisor) between any polynomials is minimized, often down to just 1. This is because when two polynomials differ by their constant, no common linear factor is shared apart from the simplest scenarios.

For example, consider the polynomials \(f_1(x) = x\) and \(f_2(x) = x + 1\). The largest polynomial that divides both is 1, because they differ consistently by an integer that ensures no shared nontrivial factor. This principle holds when expanded to any number of such polynomials with structured distinctness in their constant parts, supporting the condition of the original exercise where subsets have a nonconstant GCD.