Context: Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.

Question: What degree does not include simple formulas for cubic and quatric equations?
Answer: degree 5 and higher

Question: What concept is associated with the solvability of polynomials?
Answer: Abstract properties of Galois groups

Question: What are used to express the solutions of polynomials?
Answer: radicals

Question: What degree must it be lower than to have simple formulas?
Answer: Unanswerable

Question: What give criterion for abstract properties?
Answer: Unanswerable

Question: Which group cannot be expressed using radicals?
Answer: Unanswerable

Question: What is an example of a complex operation?
Answer: Unanswerable

Question: What do cubic and quartic equations not have?
Answer: Unanswerable