Questions on the least upper bounds and greatest lower bounds of sets of the form $\{ f(x) : x \in \mathbb{Z} \text{ or } \mathbb{R} \}$.

Refresher questions on topics in algebra that students beginning a maths undergraduate course should be familiar with.

In the Gaussian integer ring $\mathbb{Z}[i]$ , find the remainder $r=r_1+r_2i$, where $a \gt 0,\;b \gt 0$ , on dividing $a+bi$ by $c+di$ .

In the ring $\mathbb{Z}[\sqrt{2}]$ , find the remainder $r=r_1+r_2\sqrt{2}$, where $a \gt 0,\;b \gt 0$ , on dividing $a+b\sqrt{2}$ by $c+d\sqrt{2}$ .

In the  ring $\mathbb{Z}[\sqrt{-2}]$ , find the remainder $r=r_1+r_2\sqrt{-2}$, where $a \gt 0,\;b \gt 0$ , on dividing $a+b\sqrt{-2}$ by $c+d\sqrt{-2}$ .

Expanding products of 3 linear  polynomials over $\mathbb{Z}_3,\;\mathbb{Z}_5,\;\mathbb{Z}_7$

$f(X)$ and $g(X)$ are polynomials over $\mathbb{Z}_n$.

Find their greatest common divisor (GCD) and enter it as a monic polynomial.

Hence factorize $f(X)$ into irreducible factors.

Factorise 4 polynomials over $\mathbb{Z}_5$.

Given polynomial $f(X)$, $g(X)$ over $\mathbb{Q}$, find polynomials $q(X)$ and $r(X)$ over $\mathbb{Q}$ such that $f(X)=q(X)g(X)+r(X)$ and $\operatorname{deg}r(X) \lt \operatorname{deg}g(X)$.

$f(X)$ and $g(X)$ as polynomials over the rational numbers $\mathbb{Q}$.

Find their greatest common divisor (GCD) and enter as a normalized polynomial.

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Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

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