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In my abstract algebra text, the author uses "multiple" notation. Say you have a field $F$ that contains $a,b$. Consider some equation like $a^2 + 2ab + b^2 = 0$. The $2ab$ is meant to be shorthand for $ab + ab$ rather than the literal integer $2$ multiplied to $ab$.

In doing higher level computations in field theory, I encounter this notation and I'm always wondering whether or when I'm allowed to, say, divide both sides of $a^2+b^2 = -2ab$ by $2$. Can someone clarify the situations in which this multiple notation and integer multiplication coincide?

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    $\begingroup$ You can divide by $2$, if not the characteristic of the field is $2$, because the n $2=0$. $\endgroup$ – Dietrich Burde Mar 26 '16 at 20:59
  • $\begingroup$ A field $F$ contains the additive subgroup generated by $1$, so in this notation the $2$ is actually also an element of the field. If the characteristic of the field is $2$, then $2=0$; otherwise you can divide by it. $\endgroup$ – Morgan Rodgers Mar 26 '16 at 21:02
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Integers are elements in a field. For example, $1$ is the multiplicative identity, $2=1+1$, $3=1+1+1$, $4=1+1+1+1$, $5=1+1+1+1+1$, and so on. Also, $-1$ is the additive inverse of $1$, $-2$ is the additive inverse of $2$, $-3$ is the additive inverse of $3$, and so on. Therefore, you can treat them like regular elements and add, subtract, and multiply equations by integers.

However, with division, you have to be careful because you might accidentally divide by $0$. For example, in $\Bbb{Z}_2=\{0, 1\}$, $2=1+1=0$, so you can't divide by $2$ because $2=0$. This can get kind of odd, but over time, you will become wary of division, so whenever you divide by an integer, check the characteristic of the field and make sure you are not dividing by $0$.

Notice that $2ab=ab+ab$ because of the distributive property:

$$2ab=(1+1)ab=1(ab)+1(ab)=ab+ab$$

Similar logic applied for multiplying by other integers, which is why your book's "multiple" notation is consistent with this definition of an integer.

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