Field Extensions and Their Degrees · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Degrees of field extensions

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

  1. Adjoin the real cube root of 2 to the rationals. What is the degree?

    Answer: ______________

  2. Rationals with a square root of a prime adjoined. What is the extension degree?

    • 1
    • 2
    • 3
    • 4

  3. You adjoin the square root of a prime to the rationals. What is the extension degree?

    • 1
    • 2
    • 3

  4. Doubling the cube with ruler and compass is impossible since the needed extension has degree 3, not a power of 2.

    Circle one:   True   False

  5. The minimal polynomial of an algebraic number is monic and irreducible, and its degree equals the extension degree.

    Circle one:   True   False

  6. Minimal polynomial of an algebraic number is monic irreducible and its degree equals the extension degree. Is this correct?

    Circle one:   True   False

  7. Adjoin square roots of two distinct primes. What is the total degree?

    • 2
    • 3
    • 6
    • 4

  8. You adjoin square roots of two distinct primes to the rationals. What is the total degree?

    • 2
    • 3
    • 4

  9. Why can the cube root of 2 not be built by ruler and compass?

    • Its degree over rationals is 3, not a power of 2
    • It is rational
    • Square roots are impossible to build

  10. Adjoin primitive 5th roots of unity to the rationals. What is the degree?

    Answer: ______________

LightMySky · lightmysky.comW1-mt_CjOMNt5_i0-s1

Answer key

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

Degrees of field extensions W1-mt_CjOMNt5_i0-s1

  1. 3 · Minimal polynomial x cubed minus 2 is irreducible, so degree is 3.
  2. 2 · 2 is correct since the minimal polynomial is quadratic and irreducible.
  3. 2 · The minimal polynomial is quadratic and irreducible.
  4. True · Constructible degrees are powers of 2, so degree 3 blocks the build.
  5. True · The simple extension degree is defined by that polynomial.
  6. True · Degree of the simple extension is defined by that polynomial.
  7. 4 · 4 is correct by tower law: 2 times 2, while 2 misses one root and 3 and 6 mismatch the tower.
  8. 4 · The tower law gives 2 times 2, which is 4.
  9. Its degree over rationals is 3, not a power of 2 · Compass builds need power of 2 degree, and 3 does not qualify.
  10. 4 · Totient of 5 gives 4, and cyclotomic polynomial is irreducible.
Worksheet · LightMySky