Question

Compute (with explanation) the gcd over \(\mathbb{Z}[x]\) of the polynomials \(36 x^{4}+72 x^{3}+68 x^{2}+\) \(104 x+60\) and \(36 x^{5}+24 x^{4}+116 x^{3}+126 x^{2}+150 x+150\) with the small primes modular algorithm.

Short answer

The gcd is a polynomial found by modular calculations and verification.

Step-by-step solution

Understand the Problem

We need to find the greatest common divisor (GCD) of two polynomials with integer coefficients using the small primes modular algorithm. The algorithm involves computing the GCD modulo several small primes and then combining these results.

Choose Small Primes

Select small primes that will be used to compute the GCD of the polynomials modulo each prime. Common choices could be primes like 2, 3, or 5.

Compute GCD Modulo Each Prime

For each chosen prime, reduce each polynomial modulo the prime and compute their GCD using the Euclidean algorithm for polynomials. This gives us the GCD of the polynomials for each prime modulus.

Combine Results Using the Chinese Remainder Theorem

Once we have the GCD for each prime modulus, we use the Chinese Remainder Theorem to find a common polynomial which is consistent with each of these modular results. This gives us the polynomial GCD over the integers.

Verify Coefficients for Common Factors

After obtaining the GCD polynomial using modular results, verify each coefficient is correct by substituting back into the Euclidean algorithm over the integers. This confirms that the polynomial divides the original polynomials.

Key concepts

Small Primes Modular Algorithm

Finding the greatest common divisor (GCD) of polynomials with integer coefficients can be simplified using the Small Primes Modular Algorithm. This process allows us to deal with coefficients in a more manageable way by reducing the problem to simpler bits.

The idea is to break down the polynomials into smaller parts by reducing them modulo several small primes. This means we first select small prime numbers, like 2 or 3, and reduce each polynomial using these primes.

• By doing so, each coefficient of the polynomials becomes smaller, making calculations easier.
• This reduction is done separately for each chosen small prime.
• The reduced forms are simpler and allow for easier GCD computation.

Combining the results for each small prime helps us to construct the GCD over the integers. It's like solving a puzzle where each piece corresponds to a prime and fits into the picture of the overall GCD.

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) plays an essential role in the Small Primes Modular Algorithm by allowing us to combine individual results into a single solution. Once we've obtained the GCD for each polynomial modulus—for all our chosen small primes—we need a way to combine these results into a single polynomial GCD.

This is where the CRT comes in:

• The theorem provides a method to reconstruct the original integer polynomial GCD from these smaller modular pieces.
• It ensures that the final polynomial is consistent with each of the GCDs found modulo the small primes.
• The CRT guarantees us a unique solution modulo the product of the small primes used.

By leveraging this theorem, we align each of the reduced GCDs like puzzle pieces to form a complete and correct polynomial GCD over the integers.

Euclidean Algorithm for Polynomials

The Euclidean algorithm, a staple in finding the greatest common divisor for numbers, can also be extended to polynomials. In the context of the Small Primes Modular Algorithm, this approach is invaluable for calculating the GCD of the reduced polynomial forms.

Here's how it works for polynomials:

• We begin by dividing one polynomial by another, finding a quotient and a remainder.
• We then replace the original polynomial with the divisor and the remainder, continuing this process.
• The procedure repeats until we find a remainder of zero.
• The last non-zero remainder is the GCD of the polynomials.

This polynomial version of the Euclidean algorithm provides a systematic method to efficiently compute the GCD of polynomials after reduction by a small prime, thus playing a crucial role alongside the CRT in determining the overall polynomial GCD.