Fields, Rings, Groups

[congo22]

Consider all the set of all the equations for those straight lines. Create a definition for addition and multiplication such that this set and your definitions of multiplying and adding equations from a group, a ring, or a field.

explore the set under addition. is it a group?

explore the set under X is it a group?

explore the set under addition and multiply operation. Is it a ring? A communitive ring?

explore the set. is it a field.

Can someone help me come up with an example i am totally confuse...

 

[DrMike]

You are being asked to invent two ways of combining a pair of straight lines and getting a straight line. Call one of them "addition" and the other "multiplication".

Then you need to check whether these form groups. What's your identity, in each case? What are the inverses?

Your definitions can be anything you like, it seems. You can combine the lines using geometry, or using the equations for the lines, or anything at all. Just make sure that whenever you combine two lines, you get a line.

 

[congo22]

But for instant if i have x +1 and 2x +2 when i multiple i get a quardractic... so it is not in the set right? so for multiplication it is not a group? am i right?

and how would i explain why or why not it is or is not a field?

 

[daon]

But for instant if i have x +1 and 2x +2 when i multiple i get a quardractic... so it is not in the set right? so for multiplication it is not a group? am i right?

and how would i explain why or why not it is or is not a field?

Not necessarily multiplication as you know it. Define multiplication as anything you want, such as: (ax+b)?(cx+d) = (ac)x+bd. Its your space. Does the one I picked work?

 

[congo22]

but how can i explain for instance this set i made up if it a field or not?


addition is regular normal additon (ax+b)+(cx+d)=(a+c)x+(b+d)
multiplication is this(ax+b)(cx+d)=((ac)x+bd)


how would i show this set is a field or not?


Thanks

 

[daon]

Fields are commutative unitary rings (with multiplicative inverses).

1) Find a (the) unity with your given definition of multiplication (if any).
2) Show your multiplictive rule is commutative.
3) For a given line (ax+b) find a line (cx+d) such that (ax+b)(cx+d)=1.
...where 1 is your unity from (1).

Normally for 3) we'd need to check the opposite, too. i.e. (cx+d)(ax+b)=1. However, showing commutativity first takes care of this.