Let $ S= \{ i +j 2^{1/4} +k 4^{1/4} + l 8^{1/4} |i,j,k,l \in \mathbb {Z}\}$
Decide if S is a subring of the real numbers with its usual operations of addition and multiplication. if it is give a proof and if not explain why.
Well i mean this is a Ring, not every element is a unit but it is a ring that is defined for multiplication it has identity and no zero divsiors so its rather well behaved.
we dont really have theorems for sub-rings ( only groups in my intro class) how do i prove that this ring is a subring of the the field of reals?
Best Answer
Let $\alpha = 2^{1/4} \in \mathbb R$. Then, by definition, $\mathbb Z[\alpha]$ is the smallest subring of $\mathbb R$ that contains $\alpha$.
$\mathbb Z[\alpha]$ also coincides with the set of all polynomial expressions in $\alpha$ with coefficient in $\mathbb Z$.
Since $\alpha^4=2 \in \mathbb Z$, all terms of degree $4$ or higher can be replaced by lower degree terms.
Therefore, $\mathbb Z[\alpha]= \{ i +j \alpha +k \alpha^2 + l \alpha^3 : i,j,k,l \in \mathbb {Z}\}=S$, and $S$ is a ring.