Good exercises on groups, fields, rings etc I want to help out a friend with an upcoming test on structures such as groups, fields,  etc. Does anyone know good question of the form: is V,* a group? or what type of structure is V,*,# ((unital) ring, integral domain, field, etc)? Are V and Y homomorp? etc. Keep in mind that it is for a high school student. Thanks
 A: Groups: (1). Determine which of the following sets are group under addition:
(a) the set of all rationals (including $0$) in lowest term whose denominators are odd.
(b) the set of all rationals (including $0$) in lowest term whose denominators are even.
(c) the set of rationals whose absolute value $< 1.$
(d) the set of rationals whose absolute value $\geq 1$ together with $0.$
(2). Let $G:= \{a + b\sqrt 2 \mid a, b \in \mathbb Q \}.$
(a) Show that $G$ is a group under addition.
(b) Show that non-zero elements of $G$ forms a group under multiplication.
(3). Let $G:= \{z \in \mathbb C \mid z^n = 1$ for some $n \in \mathbb N \}$ (this one is little harder than the above two).
(a) Show that $G$ forms a group with respect to multiplication.
(b) Show that $G$ does not form a group with respect to addition.
A: As a very easy warm up, how about asking questions about the basic number systems?
Does $\mathbb{Z}$ form a group with respect to addition? (Yes.) Does $\mathbb{Z}$ form a group with respect to multiplication? (No.) Does $\mathbb{Q}$ form a group with respect to addition? (Yes.) Trick question: does $\mathbb{Q}$ form a group with respect to multiplication? (No, $0$ has no inverse.) Does $\mathbb{Q}\setminus \{0\}$ form a group with respect to multiplication? (Yes.) Does it form a group with respect to addition? (No, addition does not restrict to a total function on this domain.) 
