Mary Reyes

2021-12-19

Does the equation ${x}^{2}\equiv x\cdot x\equiv 2x\cdot 4xb\text{mod}7$ show that factorization of polynomials $b\text{mod}7$ is not unique? Why or why not?

Cassandra Ramirez

Beginner2021-12-20Added 30 answers

Step 1

${x}^{2}\equiv x\cdot x\equiv 2x\cdot 4xb\text{mod}7$

To show: factorization of polynomials$b\text{mod}7$ is not unique.

Other examples can be taken as

${x}^{2}\equiv 3x\cdot 5xb\text{mod}7$

${x}^{2}\equiv 6x\cdot 6xb\text{mod}7$

Step 2

From the above example it is clear that factorization of polynomials mod 7 is not unique as there are other factorization also.

To show: factorization of polynomials

Other examples can be taken as

Step 2

From the above example it is clear that factorization of polynomials mod 7 is not unique as there are other factorization also.

Orlando Paz

Beginner2021-12-21Added 42 answers

Step 1

Given that, the equation is${x}^{2}\equiv x\cdot x\equiv 2x\cdot 4x\left(b\text{mod}7\right)$ .

${x}^{2}\equiv x\cdot x\left(b\text{mod}7\right)$

${x}^{2}\equiv 2x\cdot 4x\left(b\text{mod}7\right)$

From above equation, it is observed that the factorization of the polynomials$\left(b\text{mod}7\right)$ is not unique.

Step 2

From above equation, it is observed that the factorization of the polynomials$\left(b\text{mod}7\right)$ is not unique.

For example:

${x}^{2}\equiv 2x\cdot 4x\left(b\text{mod}7\right)$

${x}^{2}\equiv (2\cdot 4)(x\cdot x)\left(b\text{mod}7\right)$

${x}^{2}\equiv (3\cdot 5)(x\cdot x)\left(b\text{mod}7\right)$

From above example, it is observed that the factorization of the polynomials$\left(b\text{mod}7\right)$ is not unique.

Because,${x}^{2}\equiv 1\left(b\text{mod}7\right)$ can have both solutions $x\equiv \pm 1\left(b\text{mod}7\right)$ . That is, $x\equiv 1\left(b\text{mod}7\right)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}x\equiv -1\left(b\text{mod}7\right)$ .

Given that, the equation is

From above equation, it is observed that the factorization of the polynomials

Step 2

From above equation, it is observed that the factorization of the polynomials

For example:

From above example, it is observed that the factorization of the polynomials

Because,

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

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