field n. a group <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo separator="true">,</mo><mo>⊕</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F,\oplus)</annotation></semantics></math>(F,⊕) with the following additional properties:
- <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo separator="true">,</mo><mo>⊕</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F,\oplus)</annotation></semantics></math>(F,⊕) is Abelian, that is, the operation <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math>⊕ is commutative: <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">∀</mi><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo>∈</mo><mi>F</mi><mo stretchy="false">)</mo><mi>f</mi><mo>⊕</mo><mi>g</mi><mo>=</mo><mi>g</mi><mo>⊕</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">(\forall f,g\in F)f\oplus g=g\oplus f</annotation></semantics></math>(∀f,g∈F)f⊕g=g⊕f: <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo separator="true">,</mo><mo>⊕</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F,\oplus)</annotation></semantics></math>(F,⊕) by itself is sometimes called the additive group of the field, and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">\mathbf 0</annotation></semantics></math>0 is the additive identity of the field.
- Another operation <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math>⊗ exists, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>∖</mo><mo stretchy="false">{</mo><mn mathvariant="bold">0</mn><mo stretchy="false">}</mo><mo separator="true">,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F\setminus\{\mathbf 0\},\otimes)</annotation></semantics></math>(F∖{0},⊗) is also a group, sometimes called the multiplicative group of the field.
- The multiplicative operation is also defined for the additive identity, although the additive identity <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">\mathbf 0</annotation></semantics></math>0 is not an element of the multiplicative group:
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">∀</mi><mi>f</mi><mo>∈</mo><mi>F</mi><mo stretchy="false">)</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>f</mi><mo>⊗</mo><mn mathvariant="bold">0</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn mathvariant="bold">0</mn><mo separator="true">;</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mn mathvariant="bold">0</mn><mo>⊗</mo><mi>f</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn mathvariant="bold">0</mn><mo separator="true">;</mo></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">(\forall f\in F)\left\{\begin{aligned} f\otimes\mathbf 0&=\mathbf 0;\\ \mathbf 0\otimes f&=\mathbf 0; \end{aligned}\right.</annotation></semantics></math>(∀f∈F){f⊗00⊗f=0;=0;
- The multiplicative operation <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math>⊗ distributes over the additive operation <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math>⊕ on either side:
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">∀</mi><mi>f</mi><mo separator="true">,</mo><mi>g</mi><mo separator="true">,</mo><mi>h</mi><mo>∈</mo><mi>F</mi><mo stretchy="false">)</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>f</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>g</mi><mo>⊕</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mi>f</mi><mo>⊗</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⊕</mo><mo stretchy="false">(</mo><mi>f</mi><mo>⊗</mo><mi>h</mi><mo stretchy="false">)</mo><mo separator="true">;</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>⊕</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>h</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mi>f</mi><mo>⊗</mo><mi>h</mi><mo stretchy="false">)</mo><mo>⊕</mo><mo stretchy="false">(</mo><mi>g</mi><mo>⊗</mo><mi>h</mi><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">(\forall f,g,h\in F)\left\{\begin{aligned} f\otimes(g\oplus h)&=(f\otimes g)\oplus(f\otimes h);\\ (f\oplus g)\otimes h&= (f\otimes h)\oplus(g\otimes h). \end{aligned}\right.</annotation></semantics></math>(∀f,g,h∈F){f⊗(g⊕h)(f⊕g)⊗h=(f⊗g)⊕(f⊗h);=(f⊗h)⊕(g⊗h).
A field must have, at a minimum, two elements, including a multiplicative identity which by definition must be distinct from its additive identity.
The Boolean field includes only two elements, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">\mathbf 0</annotation></semantics></math>0 and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn mathvariant="bold">1</mn></mrow><annotation encoding="application/x-tex">\mathbf 1</annotation></semantics></math>1:
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn mathvariant="bold">0</mn><mo>⊕</mo><mn mathvariant="bold">0</mn><mo>=</mo><mn mathvariant="bold">0</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn mathvariant="bold">0</mn><mo>⊗</mo><mn mathvariant="bold">0</mn><mo>=</mo><mn mathvariant="bold">0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn mathvariant="bold">0</mn><mo>⊕</mo><mn mathvariant="bold">1</mn><mo>=</mo><mn mathvariant="bold">1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn mathvariant="bold">0</mn><mo>⊗</mo><mn mathvariant="bold">1</mn><mo>=</mo><mn mathvariant="bold">0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn mathvariant="bold">1</mn><mo>⊕</mo><mn mathvariant="bold">0</mn><mo>=</mo><mn mathvariant="bold">1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn mathvariant="bold">1</mn><mo>⊗</mo><mn mathvariant="bold">0</mn><mo>=</mo><mn mathvariant="bold">0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn mathvariant="bold">1</mn><mo>⊕</mo><mn mathvariant="bold">1</mn><mo>=</mo><mn mathvariant="bold">0</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn mathvariant="bold">1</mn><mo>⊗</mo><mn mathvariant="bold">1</mn><mo>=</mo><mn mathvariant="bold">1</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} \mathbf 0\oplus\mathbf 0=\mathbf 0&\mathbf 0\otimes\mathbf 0=\mathbf 0\\ \mathbf 0\oplus\mathbf 1=\mathbf 1&\mathbf 0\otimes\mathbf 1=\mathbf 0\\ \mathbf 1\oplus\mathbf 0=\mathbf 1&\mathbf 1\otimes\mathbf 0=\mathbf 0\\ \mathbf 1\oplus\mathbf 1=\mathbf 0&\mathbf 1\otimes\mathbf 1=\mathbf 1\\ \end{matrix}</annotation></semantics></math>0⊕0=00⊕1=11⊕0=11⊕1=00⊗0=00⊗1=01⊗0=01⊗1=1
The additive and multiplicative operations of the Boolean field correspond to the logical bit operations
XOR
andAND
, respectively, and there is no other way to define them.