Tools of Algebra 2). Dedicated to Otto...

Algebra Level 5

$\mathbb{K}$ is a field, if and only if, the set of polynomials with coefficients in $\mathbb{K}$ , ( $\mathbb{K}[x]$ ), is an Euclidean Domain.
If $\mathbb{K}$ is a field, then $\mathbb{K}[x]$ is a field.
The polynomial $2x + 2$ is an irreducible polynomial over $\mathbb{Q}[x]$ .
The polynomial $2x + 2$ is an irreducible polynomial over $\mathbb{Z}[x]$ .

Which statement(s) above is/are true?

Clarification:

A polynomial with coefficients in an unique factorization domain $\mathbb R$ , is said to be irreducible over $\mathbb R$ if it is an irreducible element of the polynomial ring, that is, it is not invertible, not zero, and cannot be factored into the product of two non-invertible polynomials with coefficients in $\mathbb R$ .

Answer by entering the numbers of the statements you consider correct in increasing order. For example:

If you only consider the statement 3 is correct, enter 3.
If you only consider the statments 2 and 3 are correct, enter 23.
If you only consider the statements 1, 2 and 3 are correct, type 123.
If you consider all the statements are correct, type 1234.
If you consider none of the statements is correct, enter 0.

The answer is 13.

1 solution

Otto Bretscher
May 28, 2016

Thank you for a delightful problem and (and for the kind dedication), compañero! This is a wonderful refresher on topics that we learned long ago. I'm "on the road" and I don't quite have the time and leisure to write a full solution. Just a few hints:

(1) To see that $\mathbb{K}[x]$ is a Eucidean domain, observe that the degree of the polynomials is a "Euclidean Function."

(2) $x$ fails to be a unit in $\mathbb{K}[x]$

(3) If we factor $2x+2$ , then one factor will be a non-zero constant, that is, a unit in $\mathbb{Q}[x]$

(4) $2x+2=2(x+1)$ is a factorization into non-units.

Yes, that is Otto $\uparrow$ , thank you... Furthemore, it's impossible a full solution without writing a wiki... To make more entertaining "on your road" it can be proven the next one ;)

Proposition.- The next 3 statements are equivalent:

a) $\mathbb{K}$ is a field.

b) $\mathbb{K}[x]$ is an euclidean domain.

c) $\mathbb{K}[x]$ is an principal ideal domain.

Proposition.- $D$ is an (integer) domain $\iff$ $D[x]$ is an (integer) domain.

Proposition.- $D$ is an UFD $\iff$ $D[x]$ is an UFD

Guillermo Templado - 5 years ago

Tools of Algebra 2). Dedicated to Otto...

The answer is 13.

1 solution

0 pending reports