abstract algebra

• Direct Sum vs. Direct Product vs. Tensor Product
  A direct sum restricts infinite tuples to finite support, a direct product allows all tuples, and a tensor product combines spaces bilinearly.
• Find All Irreducible Polynomials of Degrees 2 and 3 in F₂[x]
  The irreducible polynomials in F₂[x] of degrees 2 and 3 are x²+x+1, x³+x+1, and x³+x²+1; a root test proves the list.
• Does Having No Roots Make a Degree-2 or Degree-3 Polynomial Irreducible?
  For polynomials of degree 2 or 3, rootlessness implies irreducibility over a field, but not over every integral domain; learn the needed conditions.
• Is the Tensor Product of a Module Element and Zero Always Zero?
  In a tensor product of modules, m ⊗ 0 and 0 ⊗ n are always zero. Bilinearity proves this, but sums of pure tensors need separate analysis.
• What Is a Quotient Ring of the Gaussian Integers?
  A quotient of the Gaussian integers depends on the ideal: learn how to form Z[i]/(α), count its elements, and determine when it is a field.
• How to Test Whether an Element of a Tensor Product Is Zero
  To test whether a tensor is zero, use a module presentation or a separating map; pure tensors and vector-space tensors have useful special criteria.