Practice: Fields and Galois theory

Fields and Galois theory

Warm-upHuman

What is the degree $[\Q(\sqrt2, \sqrt3) : \Q]$ ?

Fields and Galois theory

CoreHuman

What is the minimal polynomial of $\alpha = \sqrt2 + \sqrt3$ over $\Q$ ?

Choose one

x^4 - 10x^2 + 1

x^2 - 5

x^4 - 5x^2 + 6

(x - \sqrt2 - \sqrt3)(x + \sqrt2 + \sqrt3)

Fields and Galois theory

CoreHuman

Starting from the points $0$ and $1$ , which of these real numbers can be constructed with straightedge and compass? (Select all that apply.)

Choose all that apply

\sqrt2

\sqrt[3]{2}

\cos 72^\circ = \frac{\sqrt5 - 1}{4}

\cos 20^\circ

\sqrt{1 + \sqrt2}

\cos(2\pi/7)

Fields and Galois theory

CoreHuman

$\operatorname{Gal}(\Q(\sqrt2, \sqrt3)/\Q) = \{1, \sigma, \tau, \sigma\tau\}$ , where $\sigma$ sends $\sqrt2 \mapsto -\sqrt2$ and fixes $\sqrt3$ , and $\tau$ fixes $\sqrt2$ and sends $\sqrt3 \mapsto -\sqrt3$ . Which field is the fixed field of the subgroup $\{1, \sigma\tau\}$ ?

Choose one

\Q(\sqrt6)

\Q(\sqrt2)

\Q(\sqrt3)

\Q

Fields and Galois theory

CoreHuman

How many subfields does the field $\mathbb{F}_{2^{12}}$ (with $4096$ elements) have, including $\mathbb{F}_2$ and itself?

Fields and Galois theory

StretchHuman

Let $L$ be the splitting field of $x^4 - 2$ over $\Q$ . What is the order of $\operatorname{Gal}(L/\Q)$ ?

Fields and Galois theory

CoreHuman

Which statement correctly describes the Abel–Ruffini theorem and its Galois-theoretic refinement?

Choose one

No polynomial of degree

5

has roots expressible by radicals.There is no formula in radicals for the roots of the general quintic; a specific polynomial is solvable by radicals exactly when its Galois group is solvable, and

x^5 - 4x + 2

, with Galois group

S_5

, is not.Some quintics have no complex roots.The roots of a quintic cannot be computed numerically.

Fields and Galois theory

Warm-upHuman

Which statement about rings with four elements is correct?

Choose one

\Z/4\Z

is not a field, because

2 \cdot 2 = 0

; the field with four elements is

\mathbb{F}_4 = \mathbb{F}_2[x]/(x^2 + x + 1)

, which has characteristic

2

.

\Z/4\Z

is the field with four elements.There is no field with four elements, since

4

is not prime.

\Z/2\Z \times \Z/2\Z

(componentwise operations) is the field with four elements.

Fields and Galois theory

CoreFormal

For fields F, K, L with [Algebra F K] [Algebra K L] [Algebra F L] [IsScalarTower F K L], which Lean statement is the tower law?

Choose one

finrank F K * finrank K L = finrank F Lfinrank F K + finrank K L = finrank F Lfinrank K F * finrank L K = finrank L Ffinrank F L = finrank F K ^ finrank K L

Fields and Galois theory

StretchComputational

Write count_irreducible(p, n) returning the number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_p$ , using Gauss's formula $N(n) = \frac1n \sum_{d \mid n} \mu(d)\, p^{n/d}$ . You will need the Möbius function $\mu$ : $\mu(d) = 0$ if $d$ has a square factor, otherwise $(-1)^{k}$ where $k$ is the number of prime factors of $d$ .

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