Editing Splitting field (section)

=== The complex numbers ===
Consider the [[polynomial ring]] '''R'''[''x''], and the [[irreducible polynomial]] {{nowrap|1=''x''<sup>2</sup> + 1.}} The [[quotient ring]] {{nowrap|1='''R'''[''x'']&thinsp;/&thinsp;(''x''<sup>2</sup> + 1)}} is given by the [[Congruence relation|congruence]] {{nowrap|1=''x''<sup>2</sup> ≡ −1.}} As a result, the elements (or [[equivalence class]]es) of {{nowrap|1='''R'''[''x'']&thinsp;/&thinsp;(''x''<sup>2</sup> + 1)}} are of the form {{nowrap|1=''a'' + ''bx''}} where ''a'' and ''b'' belong to '''R'''. To see this, note that since {{nowrap|1=''x''<sup>2</sup> ≡ −1}} it follows that {{nowrap|1=''x''<sup>3</sup> ≡ −''x''}}, {{nowrap|1=''x''<sup>4</sup> ≡ 1}}, {{nowrap|1=''x''<sup>5</sup> ≡ ''x''}}, etc.; and so, for example {{nowrap|1=''p'' + ''qx'' + ''rx''<sup>2</sup> + ''sx''<sup>3</sup> ≡ ''p'' + ''qx'' + ''r''(−1) + ''s''(−''x'') = (''p'' − ''r'') + (''q'' − ''s'')''x''.}}

The addition and multiplication operations are given by firstly using ordinary polynomial addition and multiplication, but then reducing modulo {{nowrap|1=''x''<sup>2</sup> + 1}}, i.e. using the fact that {{nowrap|1=''x''<sup>2</sup> ≡ −1}}, {{nowrap|1=''x''<sup>3</sup> ≡ −''x''}}, {{nowrap|1=''x''<sup>4</sup> ≡ 1}}, {{nowrap|1=''x''<sup>5</sup> ≡ ''x''}}, etc. Thus:
:<math>(a_1 + b_1x) + (a_2 + b_2x) = (a_1 + a_2) + (b_1 + b_2)x, </math>
:<math>(a_1 + b_1x)(a_2 + b_2x) = a_1a_2 + (a_1b_2 + b_1a_2)x + (b_1b_2)x^2 \equiv (a_1a_2 - b_1b_2) + (a_1b_2 + b_1a_2)x \, . </math>
If we identify {{nowrap|1=''a'' + ''bx''}} with (''a'',''b'') then we see that addition and multiplication are given by
:<math>(a_1,b_1) + (a_2,b_2) = (a_1 + a_2,b_1 + b_2), </math>
:<math>(a_1,b_1)\cdot (a_2,b_2) = (a_1a_2 - b_1b_2,a_1b_2 + b_1a_2). </math>

We claim that, as a field, the quotient ring {{nowrap|1='''R'''[''x''] / (''x''<sup>2</sup> + 1)}} is [[isomorphic]] to the [[complex number]]s, '''C'''. A general complex number is of the form {{nowrap|1=''a'' + ''bi''}}, where ''a'' and ''b'' are real numbers and {{nowrap|1=''i''<sup>2</sup> = −1.}} Addition and multiplication are given by

:<math>(a_1 + b_1 i) + (a_2 + b_2 i) = (a_1 + a_2) + i(b_1 + b_2),</math>
:<math>(a_1 + b_1 i) \cdot (a_2 + b_2 i) = (a_1a_2 - b_1b_2) + i(a_1b_2 + a_2b_1).</math>

If we identify {{nowrap|1=''a'' + ''bi''}} with (''a'', ''b'') then we see that addition and multiplication are given by

:<math>(a_1,b_1) + (a_2,b_2) = (a_1 + a_2,b_1 + b_2),</math>
:<math>(a_1,b_1)\cdot (a_2,b_2) = (a_1a_2 - b_1b_2,a_1b_2 + b_1a_2).</math>

The previous calculations show that addition and multiplication behave the same way in {{nowrap|1='''R'''[''x'']&thinsp;/&thinsp;(''x''<sup>2</sup> + 1)}} and '''C'''. In fact, we see that the map between {{nowrap|1='''R'''[''x'']&thinsp;/&thinsp;(''x''<sup>2</sup> + 1)}} and '''C''' given by {{nowrap|1=''a'' + ''bx'' → ''a'' + ''bi''}} is a [[homomorphism]] with respect to addition ''and'' multiplication. It is also obvious that the map {{nowrap|1=''a'' + ''bx'' → ''a'' + ''bi''}} is both [[injective]] and [[surjective]]; meaning that {{nowrap|1=''a'' + ''bx'' → ''a'' + ''bi''}} is a [[bijective]] homomorphism, i.e., an [[ring isomorphism|isomorphism]]. It follows that, as claimed: {{nowrap|1='''R'''[''x'']&thinsp;/&thinsp;(''x''<sup>2</sup> + 1) ≅ '''C'''.}}

In 1847, [[Augustin-Louis Cauchy|Cauchy]] used this approach to ''define'' the complex numbers.<ref>{{Citation|last = Cauchy|first = Augustin-Louis|author-link = Augustin-Louis Cauchy|title = Mémoire sur la théorie des équivalences algébriques, substituée à la théorie des imaginaires|journal = [[Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences]]|volume = 24|year = 1847|language = fr|pages = 1120–1130}}</ref>