$(\mathbb R, \oplus)$ is a group. Define a multiplication with which we get a field. Where $a \oplus b = a + b +1$

Solution 1:

If you consider $a\otimes b= ab+a+b$. Is clear that is a commutative operation because of the structure. Notice that:

$$(a\otimes b) \oplus (a\otimes c)= (ab+a+b)\oplus (ac+a+c)=(ab+a+b)+(ac+a+c)+1$$ $$=a(b+c)+2a+b+c+1$$ That is equal to $$a\otimes(b\oplus c)=a\otimes(b+c+1)=a(b+c+1)+a+(b+c+1)$$ $$=a(b+c)+2a+b+c+1$$

Finally look that $$a\otimes(b\otimes c)=a\otimes (bc+b+c)=a(bc+b+c)+a+(bc+b+c)$$ $$=abc+ab+ac+bc+a+b+c$$ And $$(a\otimes b)\otimes c=(ab+a+b)\otimes c=(ab+a+b)c+(ab+a+b)+c$$ $$=abc+ab+ac+bc+a+b+c$$

Solution 2:

The most straightforward approach to solving this problem is to prove $(\mathbb{R}, \oplus) \cong (\mathbb{R}, +)$. Once you've done that, you can use the isomorphism to define $\otimes$ in terms of $\times$.