The natural numbers \(\mathbb{N}\) are a field.

• True
• False

The integers \(\mathbb{Z}\) are a field.

• True
• False

The set \(\{p + \mathrm i q | p,q\in\mathbb Q\}\subset \mathbb C\) equipped with complex addition and multiplication is a field.

• True
• False

Let \(\mathbb F\) be a field and \(x,y\in\mathbb F.\) If \(xy=0,\) then \(x=0\) or \(y=0.\)

• True
• False

Given fields \(\mathbb{F}\) and \(\mathbb{K},\) The zero function \(o :\mathbb{F}\rightarrow\mathbb{K},\) which sends all elements of \(\mathbb{F}\) to \(0_{\mathbb{K}},\) is a field homomorphism.

• True
• False

If \(\chi:\mathbb F\to\mathbb K\) is a field homomorphism, then \(\chi(x)=0_{\mathbb K}\) implies \(x=0_{\mathbb F}.\)

• True
• False

There is a field homomorphism from \(\mathbb{F}_4\) to \(\mathbb{F}_2.\)

• True
• False

The least value of \(n\in \mathbb N\) such that \(\operatorname{Im}((1+\mathrm i\sqrt 3)^n)=0\) equals \(n=3.\)

• True
• False

\(\operatorname{Im}(z)+\mathrm i \operatorname{Re}(z) = \mathrm i \bar z\) for all \(z\in\mathbb C.\)

• True
• False

\(\mathrm i^{1291}=\mathrm i.\)

• True
• False

If \(z\in\mathbb{C}\) is such that \(\operatorname{Re}\left(\frac{z-1}{z+1}\right)=0,\) then \(|z|=1.\)

• True
• False

If \(z\in\mathbb{C}\) is such that \(\operatorname{Im}\left(\frac{z-1}{z+1}\right)=0,\) then \(\operatorname{Im}(z) = 0.\)

• True
• False

If \(z=(\mathrm i-\mathrm i^2)^3,\) then \(\bar z=2+2\mathrm i.\)

• True
• False

A complex number \(z\) is purely imaginary if and only if \(\bar z = -z.\)

• True
• False

If \(n\) is an odd natural number, then \(\mathrm i^2+\mathrm i^4 + \ldots +\mathrm i^{2n} = 0.\)

• True
• False