The Galois Correspondence · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Symmetries of fields and the groups that mirror them

Mathematics · Abstract Algebra · ages 22-24
Name ______________________   Date ____________

  1. Splitting field of a quadratic with distinct roots over the rationals. What is the Galois group order?

    Answer: ______________

  2. Rationals with a square root of a prime adjoined. How many automorphisms fix the rationals?

    • 1
    • 4
    • 2
    • 3

  3. Adjoin the square root of a prime to the rationals. How many automorphisms fix the rationals?

    • 4
    • 3
    • 2

  4. The general quintic is unsolvable by radicals because its Galois group S5 is not solvable.

    Circle one:   True   False

  5. Adjoin two independent square roots to the rationals. What is the order of the Galois group?

    Answer: ______________

  6. Galois correspondence reverses inclusions: larger subgroups correspond to smaller intermediate fields. Is this correct?

    Circle one:   True   False

  7. Which fields lie strictly between the rationals and the compositum of square roots of 2 and 3?

    • Rationals with sqrt2, with sqrt3, with sqrt6
    • Rationals with cbrt2 only
    • No intermediate fields
    • Rationals with pi

  8. Which fields lie strictly between the rationals and the rationals with roots 2 and 3 adjoined?

    • Rationals with root 2, with root 3, with root 6
    • Rationals with cube root 2 only
    • No fields lie strictly between

  9. The splitting field of x cubed minus 2 has how many automorphisms fixing the rationals?

    • 3
    • 6
    • 9

  10. Symmetric group on 3 letters has how many subgroups in total?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Symmetries of fields and the groups that mirror them W1-mt_30kyliQR_L-s1

  1. 2 · Degree is 2, so Galois group has 2 elements.
  2. 2 · 2 is correct: identity and sign change on the root, while 1 misses conjugation and 4 and 3 overcount.
  3. 2 · Only the identity and the sign change on the root work, so there are 2.
  4. True · A non solvable group blocks every expression of the roots by radicals.
  5. 4 · The degree is 2 times 2, and the group order equals the degree.
  6. True · Fixed fields shrink as groups grow.
  7. Rationals with sqrt2, with sqrt3, with sqrt6 · Rationals with sqrt2, with sqrt3, with sqrt6 is correct for the Klein four case, while the others miss quadratics or add transcendental elements.
  8. Rationals with root 2, with root 3, with root 6 · The Klein four case yields exactly three quadratic subfields.
  9. 6 · The splitting degree is 6 with full symmetric group S3.
  10. 6 · Subgroups are trivial, three of order 2, one of order 3, whole group, totaling 6.
Worksheet · LightMySky