Sample Questions

1
Sample Questions

• Who found the solution to the cubic equation?
  When was this done?
• Why were there cases historically and who solved
  which one.
• Who solved the quartic equation? When was this
  done?
• What is the algorithm for an equation of order 4?
• Who showed that there can be no general algebraic
  solution to the quintic equation?
• Who proved that there is no general solution to
  any higher order equation (ngt4)?
• What is the controversy between Cardano and
  Tartaglia?

• What was Gauss contribution to the solution of
  algebraic equations?
• What is the restriction in Cardanos original
  work for the use of his formula? How did this
  historically develop?
• Where did algebra come from?
• How did the Greeks treat algebraic equations?
• Give the rule for the solution of the cosa!
• Prove the rule!
• Give Lagranges general solution to a cubic
  equation.
• What is a Lagrange resolvent?
• What was Lagranges approach to solving algebraic
  equations.

2
Study Guide 5

• What are the reduced and the proposed equations?
• Express the roots of the reduced equation in
  terms of the roots of the proposed equation!
• How does Lagrange argue that the reduced equation
  must be of degree six?
• What is Lagranges general approach to solving
  equations?
• Show that the coefficients of the proposed
  equation are symmetric in the roots of that
  equation!
• What is the contribution of Galois to the theory
  of algebraic solutions of equations? When did he
  live and work? How did he die?
• When and how was his work published?

• What is a Galois resolvent?
• What is a splitting field?
• How is the spitting field related to the
  resolvent?
• What is the approach of Galois to distinguish
  algebraically solvable from algebraically
  unsolvable equations?
• What were his main new ideas?
• State the fundamental theorem of algebra!
• How did Galois show that there are no algebraic
  formulas for higher degree equations (ngt5)?
• How are his results related to the results of
  Gauss?