Given a field F, Quillen defined K-groups to be the higher homotopy groups of a space . Though apparently the space itself can be easily and explicitly constructed, the K-groups are mysterious gadgets. However the *homology groups *of the connected component of K(F) containing its base point coincide with the homology groups of the general linear group.

Anyway, back to Quillen K-groups. For , these groups coincide with the Milnor K groups which are more algebraic in nature and much easier to define. The n-th Milnor K group of a field F is defined as follows :

Let denote the abelian group of non-zero elements of the field F. Then

where is the subgroup generated by tensors where for some , . The class of the tensor in is denoted .

Thus for , we have . These are the the so-called *Steinberg* relations.

Thus, we have

Let us do some symbolic computations in to get a feel for this object (and also because it is fun!)

**The Rules**

- Since is bilinear, .
- Similarly .

So which implies . Similarly . Using bilinearity repeatedly, .

If , maybe this needs some further explanation. For instance if say, then ..

**The Game**

I. Show .

This is because .

II. Show .

This is because .

III. Show .

This is because .

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