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I have some basic question with polynomials appreciate if someone could explain me this.

Following is additional and multiplication tables and it is say that this is a field. Have no idea why say it is field.

Please explain me.

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Do you know what a field is? –  Rhys Apr 30 '13 at 15:23
    
@Rhys- I know very less. The definitions are very difficult to understand me. –  New Developer Apr 30 '13 at 15:25
    
See the answers below to understand what a field is. Even if the notations are a bit complicated for you to understand, you should get the hang of what the answerers explain to you. –  NasuSama Apr 30 '13 at 15:27

2 Answers 2

up vote 13 down vote accepted

Just note that, for multiplication, every non-zero element on the first column has, on its line, a $1$, which means it has a inverse to the right... Now, for every element on the first row, which contains nonzero elements, it has a 1 on its column, which means it is left-invertible.

Thus, it is a field since every nonzero element is invertible.

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You made a good point! –  NasuSama Apr 30 '13 at 15:25
    
And I just entered Math Stack Exchange! :P Begginer's luck I guess. –  OhMyGod Apr 30 '13 at 15:25
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Don't worry about this. Everyone has the first time. Nobody can go far beyond the "StackExchange" habitants, or people who go on that site for very long time (Know: experience!) Good job posting the great answer! –  NasuSama Apr 30 '13 at 15:26
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Last line must be: "...since every nonzero element is invertible"...just nitpicking, and welcome! +1 –  DonAntonio Apr 30 '13 at 15:32
    
In order to become a field, the two scenarios (row wise and column wise) should be there or having 1 is enough? –  New Developer Apr 30 '13 at 15:33

If $R$ is a ring with unity and $f\in R[x]$ a polynomial, then $R[x]/\langle f\rangle$ is always at least a ring. To be a field, it needs a unity (check) and each nonzero element must have a multiplicative inverse (check).

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