Okay so I'm working on this problem:
The function $\phi:\mathbb{C}→\mathbb{C}$ given by $\phi(z)=\overline{z}$ is a ring isomorphism (where $\overline{z}$ is the complex conjugate of $z$).
I'm still very new to abstract algebra, so maybe it's obvious, but here's what I have: A set is a ring is it has two binary operations - addition and multiplication - closed and satisfying commutativity, associativity, distributive, A zero or identity element, and an inverse element. A function is an isomorphism if it forms a bijection between the two spaces - or rings in this case - meaning it is surjective and injective (1-1 and onto).
The complex field is a ring, given that it is closed under multiplication and addition, has the distributive, commutative, and associative properties, has a zero element (0), and has inverses - I believe in this case the conjugate.
Now, all I need to do is determine if the function $\phi$ is an isomorphism. I think it is, but I'm not sure how to go about deciding for certain. Any help or pointers would be appreciated, or if you see that I've done something wrong in saying $\mathbb{C}$ is a ring. Thanks!
You would have to verify the definition:
Is $\phi$ a ring homomorphism? What does this mean? What do you have to verify?
Is $\phi$ a bijection. Is it one-one and onto? Is there an easy inverse of it that you can see? (As Liddo pointed out in the comments, what is $\phi(\phi(z))$?)