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Galois Theory

Test your understanding of Galois Theory, a branch of abstract algebra that studies field extensions and their automorphisms.

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is the fundamental theorem of Galois theory?

  1. Every finite extension of a field is normal.
  2. Every finite normal extension of a field is separable.
  3. The Galois group of a finite extension of a field is isomorphic to the group of automorphisms of the extension.
  4. All of the above.
Question 2 Multiple Choice (Single Answer)

What is a Galois extension?

  1. A finite extension of a field that is normal and separable.
  2. A finite extension of a field that is normal but not necessarily separable.
  3. A finite extension of a field that is separable but not necessarily normal.
  4. A finite extension of a field that is neither normal nor separable.
Question 3 Multiple Choice (Single Answer)

What is the Galois group of a field extension?

  1. The group of all automorphisms of the extension.
  2. The group of all automorphisms of the extension that fix the base field.
  3. The group of all automorphisms of the extension that fix the base field and its subfields.
  4. The group of all automorphisms of the extension that fix the base field and its subfields and their subfields.
Question 4 Multiple Choice (Single Answer)

What is the order of the Galois group of a Galois extension?

  1. The degree of the extension.
  2. The number of elements in the extension.
  3. The number of automorphisms of the extension.
  4. The number of automorphisms of the extension that fix the base field.
Question 5 Multiple Choice (Single Answer)

What is the solvability of a polynomial?

  1. The ability to express its roots in terms of radicals.
  2. The ability to express its roots in terms of elementary functions.
  3. The ability to express its roots in terms of algebraic functions.
  4. The ability to express its roots in terms of transcendental functions.
Question 6 Multiple Choice (Single Answer)

What is the Galois group of a polynomial?

  1. The group of all automorphisms of the splitting field of the polynomial.
  2. The group of all automorphisms of the splitting field of the polynomial that fix the base field.
  3. The group of all automorphisms of the splitting field of the polynomial that fix the base field and its subfields.
  4. The group of all automorphisms of the splitting field of the polynomial that fix the base field and its subfields and their subfields.
Question 7 Multiple Choice (Single Answer)

What is the irreducibility of a polynomial?

  1. The inability to factor the polynomial into a product of two non-constant polynomials.
  2. The inability to factor the polynomial into a product of two non-linear polynomials.
  3. The inability to factor the polynomial into a product of two non-quadratic polynomials.
  4. The inability to factor the polynomial into a product of two non-cubic polynomials.
Question 8 Multiple Choice (Single Answer)

What is the Eisenstein criterion for irreducibility?

  1. If a polynomial has an integer coefficient, a leading coefficient of 1, and a constant term that is not divisible by the leading coefficient, then it is irreducible over the integers.
  2. If a polynomial has a rational coefficient, a leading coefficient of 1, and a constant term that is not divisible by the leading coefficient, then it is irreducible over the rationals.
  3. If a polynomial has a real coefficient, a leading coefficient of 1, and a constant term that is not divisible by the leading coefficient, then it is irreducible over the reals.
  4. If a polynomial has a complex coefficient, a leading coefficient of 1, and a constant term that is not divisible by the leading coefficient, then it is irreducible over the complex numbers.
Question 9 Multiple Choice (Single Answer)

What is the primitive element theorem?

  1. Every finite field extension has a primitive element.
  2. Every finite Galois extension has a primitive element.
  3. Every finite normal extension has a primitive element.
  4. Every finite separable extension has a primitive element.
Question 10 Multiple Choice (Single Answer)

What is the Kronecker-Weber theorem?

  1. Every abelian extension of the rationals is contained in a cyclotomic field.
  2. Every abelian extension of the rationals is contained in a quadratic field.
  3. Every abelian extension of the rationals is contained in a cubic field.
  4. Every abelian extension of the rationals is contained in a quartic field.
Question 11 Multiple Choice (Single Answer)

What is the Hilbert irreducibility theorem?

  1. Every polynomial with integer coefficients that is irreducible over the rationals is also irreducible over the integers.
  2. Every polynomial with rational coefficients that is irreducible over the rationals is also irreducible over the integers.
  3. Every polynomial with real coefficients that is irreducible over the rationals is also irreducible over the integers.
  4. Every polynomial with complex coefficients that is irreducible over the rationals is also irreducible over the integers.
Question 12 Multiple Choice (Single Answer)

What is the Artin-Whaples theorem?

  1. Every finite Galois extension of the rationals is solvable.
  2. Every finite Galois extension of the rationals is solvable by radicals.
  3. Every finite Galois extension of the rationals is solvable by elementary functions.
  4. Every finite Galois extension of the rationals is solvable by algebraic functions.
Question 13 Multiple Choice (Single Answer)

What is the Galois correspondence?

  1. A bijection between the subgroups of the Galois group of a Galois extension and the subfields of the extension.
  2. A bijection between the subgroups of the Galois group of a Galois extension and the quotient fields of the extension.
  3. A bijection between the subgroups of the Galois group of a Galois extension and the subrings of the extension.
  4. A bijection between the subgroups of the Galois group of a Galois extension and the ideals of the extension.
Question 14 Multiple Choice (Single Answer)

What is the inverse Galois problem?

  1. Given a group, construct a Galois extension whose Galois group is isomorphic to the given group.
  2. Given a group, construct a field extension whose Galois group is isomorphic to the given group.
  3. Given a group, construct a ring extension whose Galois group is isomorphic to the given group.
  4. Given a group, construct a module extension whose Galois group is isomorphic to the given group.
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