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Groups, rings, fields all seem similar to me. I don't get the difference between them.

What is the difference between them algebraically and graphically?
Please give suitable example.

Jessica Tiberio
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Love Invariants
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    "Graphically"? I think you better grab a book in abstract algebra and read there. – DonAntonio May 18 '18 at 08:30
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    To begin, a group only has one operation. Rings and fields have 2 operations. Fields are commutative and every element has an inverse with the exception of the multiplicative identity. Rings need not be commutative, elements need not have multiplicative inverses – pureundergrad May 18 '18 at 08:30
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    "Groups, rings, fields all seem similar to me..." Really?? On a group there is a multiplication, on a ring a multiplication and an addition, a field is a ring with a special property. But as @DonAntonio says. Just find a good source that gives you more details. – drhab May 18 '18 at 08:30
  • Sadly, this duplicates a post from 8 years ago. Please use the search feature first next time. This question I offered as a duplicate showed up for me as the top "related" question on the right when I first opened this question. – rschwieb May 18 '18 at 13:14
  • @rschwieb-I saw that post but that doesn't give any geometrical insight. – Love Invariants May 18 '18 at 15:37
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    @LoveInvariants I'm afraid that attempt at distinction does not really hold up. There is no clear "geometrical" (or "graphical" either, for that matter) distinction and besides, the answer you accepted made no attempt to explain anything geometrically. So apparently that aspect was not important. – rschwieb May 18 '18 at 20:09
  • @MorganRodgers-Graphical insight – Love Invariants May 20 '18 at 06:33
  • @MorganRodgers-On some paper I read this.• Looking at the roots ◦ Bound their location on the complex plane. ◦ Examine the algebraic degree of the roots, and consider field extensions. Minimal polynomials. – Love Invariants May 20 '18 at 06:35

1 Answers1

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Groups, rings, fields all seem similar to me. I don't get the difference between them.

Similar in what sense? The difference between a group and a ring is fundamental: a group is a pair $(G, \circ)$ with some additional axioms. A ring on the other hand is a triple $(R, +, \cdot)$ such that $(R,+)$ is an (abelian) group and some additonal axioms on $\cdot$.

Fundamentally a group has only one binary operation while a ring is a set equipped with two binary operations.

For example the set $\{-1, 1\}$ together with usual number multiplication is a group. Another example is a group of all permutations over a finite set $S_n$. On the other hand integers $\mathbb{Z}$ together with usual addition and multiplication is a ring. Note that this implies that $\mathbb{Z}$ is also a group if you look only at the addition.

Now a field is just a special ring: the one where every non-zero element is invertible (with respect to the second binary operation). For example $\mathbb{Z}$ is not a field, because $2$ has no multiplicative inverse (being $\frac{1}{2}$) but the set of rationals $\mathbb{Q}$ is a field. You can also think about fields as rings where division (by non-zero elements) is well defined.

To fully understand these definitions you need to read some sources. Wikipedia is a good place to start:

https://en.wikipedia.org/wiki/Group_(mathematics) https://en.wikipedia.org/wiki/Ring_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics)

freakish
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