1. ## Integral domain problems

Hi,

I've a couple of problems that are driving me crazy:

Prove or disprove that if a,b ε D and a > b, then a² > b².

Prove or disprove that if a,b ε D and a > b, then a³ > b³.

Should I start with a² > b² and try and disprove it for a²=b², and a² < b²? I have been trying it that way but no success as of yet.

Any help would be much appreciated.

2. If a and b are both positive and $a > b$, then $ab > b^2$ (multiplying the inequality by b on both sides) and $a^2 > ab$ (multiplying the inequality by a on both sides). Then by transitivity, $a^2 > ab > b^2$ implies $a^2 > b^2$. The proof that $a^3 > b^3$ is similar.

3. Originally Posted by jackiemoon
Hi,

I've a couple of problems that are driving me crazy:

Prove or disprove that if a,b ε D and a > b, then a² > b².

Prove or disprove that if a,b ε D and a > b, then a³ > b³.

Should I start with a² > b² and try and disprove it for a²=b², and a² < b²? I have been trying it that way but no success as of yet.

Any help would be much appreciated.
ok, that's very confusing because you obviously haven't posted all the question! how's the order < or > defined over your integral domain??

4. Hey thanks for the quick reply. I understand what you are saying but we are told a,b ε D, not Dp , so they could be either pos. or neg. or zero.

5. Originally Posted by jackiemoon

Hey thanks for the quick reply. I understand what you are saying but we are told a,b ε D, not Dp , so they could be either pos. or neg. or zero.
there's no such thing as postive or negative in an integral domain in general! your question makes absolutely no sense! there's a very important piece of information missing in your question!