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

Algebra Level 5
1. $$\mathbb{K}$$ is a field, if and only if, the set of polynomials with coefficients in $$\mathbb{K}$$, ($$\mathbb{K}[x]$$), is an Euclidean Domain.
2. If $$\mathbb{K}$$ is a field, then $$\mathbb{K}[x]$$ is a field.
3. The polynomial $$2x + 2$$ is an irreducible polynomial over $$\mathbb{Q}[x]$$.
4. 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.
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