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So I just started on abstract algebra and I have a few amateur question which I hope to clarify.

So basically is subring $R$ of a field $F$ always communtative ? ($R$ contains $0$ and $1$)

If no, can I have a counter-example.

If yes, how do I go about proving it ?

Any help or insights is deeply appreciated.

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    What is the definiton os a field? – 2017-01-16
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    What can you say about a subring of a commutative ring? – 2017-01-16
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    Intuitively, I would expect the communtative property to be preserved. Since what-ever element in $R$ is in $F$, but is this a good enough argument ? – 2017-01-16
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    If F is commutative and R is a subring, then $\forall x, y \in R$ we have $x, y\in F $ so they commute. However, as @Mathchat said, what is the definiton of a field? Does it imply commutativity? – 2017-01-16
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    One of the field axioms is communtativity of multiplication. I see your point now. Thank you for taking the time to answer my question. – 2017-01-16
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    Hm, interesting. I always thought that the french structure named "corps" would translate into "field", but actually "field" translates into "corps commutatif", as a field does require commutativity. Today I Learned. – 2017-01-16

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A subring of any commutative ring (in particular, of a field) is commutative. Indeed, if $R$ is a commutative ring and $S\subseteq R$ is a subring, then multiplication of elements of $S$ is the same operation as their multiplication in $R$. So if $x,y\in S$, then $xy=yx$ in $R$ since $R$ is commutative, and hence $xy=yx$ in $S$ as well.