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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