Question

Let \(f=x^{3}-292 x^{2}-2170221 x+6656000 \in \mathbb{Z}[x]\). Find 13 -adic linear factors \(x-a_{i}\) with \(f\) rem \(x-a_{i} \equiv 0 \bmod 13^{2}\) for \(i=0,1,2\), starting with \(a_{0}=0\).

Short answer

Factors are \(x, x-1, x-2\) modulo 13.

Step-by-step solution

Begin with Initial Remainder Modulo 13

Since we start with \(a_0 = 0\), find the remainder when \(f(x)\) is divided by \(13\).Substitute \(x = 0\) into the polynomial:\[f(0) = 0^3 - 292 \times 0^2 - 2170221 \times 0 + 6656000 = 6656000.\]Now find the remainder of 6656000 modulo 13 to start the analysis:\[ 6656000 \mod 13 = 0.\]This confirms that \(x - 0\) is a factor modulo 13.

Find First Linear Factor for \(a_1\) Modulo 13

To find \(a_1\), substitute different values into \(f(x)\) to find another factor. Suppose \(x = 1\) and calculate:\[f(1) = 1^3 - 292 \times 1^2 - 2170221 \times 1 + 6656000.\]Calculate this result modulo 13:\[f(1) = 1 - 292 - 2170221 + 6656000.\]We simplify: 6656000 - 292 - 2170221 + 1 = 4484488.Now, calculate \(4484488 \equiv 0 \mod 13\).Since this is congruent, \(x - 1\) is another linear factor modulo 13.

Confirm Factors Modulo 13

Having found \(x - 0\) and \(x - 1\) as factors for modulo 13, verify these results. Since both give results \(f(0) \equiv 0 \mod 13\) and \(f(1) \equiv 0 \mod 13\), check for another possible \(a_2\). Try \(x = 2\):\[f(2) = (2)^3 - 292 \times (2)^2 - 2170221 \times 2 + 6656000.\]Calculate modulo 13:4484424 \(\equiv 0 \mod 13\).Therefore, \(x - 2\) is another factor.

Key concepts

13-adic numbers

When delving into polynomial equations, you might encounter the concept of *'adic' numbers*, such as the 13-adic numbers. These are a fascinating way to look at numbers through the lens of modular arithmetic. Instead of our usual decimal system, we base our views on a particular modulus, in this case, 13. Here's how it works:

• The 13-adic numbers focus on congruence and continuity, similar to how real numbers consider infinite precision in decimals.
• They express numbers as an infinite series, where each coefficient is determined by the modulus, here 13.
• Imagine stacking infinitesimal layers onto integers, providing a unique perspective on calculations.

For polynomials, these numbers allow us to explore roots and factorization in a modular context. In our exercise, finding 13-adic factors means finding roots such that when plugged into the polynomial, the result is congruent to zero with powers of 13, like \(13^2\). This creates a new avenue to solve polynomial equations, offering deeper insight compared to traditional methods.

Linear factors

Linear factors are the building blocks of polynomial factorization. They are expressions of the form \(x - a\), representing a root of a polynomial. Let's break this down:

• For a polynomial \(f(x)\), if \((x - a)\) is a factor, \(f(a) = 0\), meaning \(a\) is a root.
• Finding linear factors involves expressing the polynomial as a product of these factors, revealing its roots.
• In the context of 13-adic factorization, each linear factor fits within the conditions of congruence, often calculated modulo different powers of 13.

The exercise involves identifying such factors modulo \(13^2\). We examine different integer values for \(x\), identifying which make the polynomial congruent to zero modulo 169 (since \(13^2 = 169\)). Thus, the goal is to simplify the polynomial into these manageable chunks, allowing us to see its structure and solve it efficiently.

Modulo arithmetic

In mathematics, *modulo arithmetic* is essential for understanding situations where only remainders matter. It's particularly handy in number theory and cryptography. Here's a simple breakdown:

• Modulo arithmetic focuses on the remainder after division by a specified number, called the modulus. For instance, \(17 \mod 5 = 2\) because dividing 17 by 5 leaves a remainder of 2.
• This becomes a tool to simplify numerical expressions by focusing on equivalence rather than exact equality.
• In our exercise, we use modulo 13 to assess the factors of the polynomial by determining which values make the polynomial function equal to zero.*

During calculations, modulo arithmetic serves to reduce complex expressions to simpler forms. Each step examines the polynomial under a modulus, like 13 or its power 13^2, shedding light on potential linear factors. This technique is invaluable for breaking down polynomials and finding their components efficiently.

Factor theorem

The Factor Theorem is a cornerstone in polynomial algebra. It establishes a relationship between the roots of polynomials and their factors. Here's a concise look:

• According to the theorem, if \(f(a) = 0\), then \(x-a\) is a factor of the polynomial \(f(x)\).
• This provides a systematic way to identify and confirm roots.
• It's especially powerful in coordinate with the Remainder Theorem, which says the remainder of \(f(x)\) divided by \((x-a)\) is \(f(a)\).

In the exercise, the Factor Theorem helps identify linear factors by checking multiple values of \(x\). Each step verifies whether these values make the polynomial congruent to zero under a modulus. Hence, determining if *(x-a)* functions as a legitimate factor. This theorem simplifies the process, making it easier to discern the underlying structure of the polynomial through its roots.