/sci/ - Science & Math

>>6020907

Let <span class="math">F = K(u_1, u_2, ..., u_n)[/spoiler] be a field extension of K.

Why isn't [F : K] = n? Doesn't {u_i} form a basis for F as a vector space over K? In general, is there a way of determining [F : K] if F/K is given as above?

One summer, an engineer, a physicist, and a mathematician were vacationing in England. Wanting to see the countryside, they boarded a train and were staring out the window when the train passed a single sheep grazing in the field.

The engineer promptly remarked, "I see the sheep in England are black."

To which the physicist replied, "No, sir. What we know is that this single sheep is black."

To which the mathematician angrily corrected, "No, you fools! What we know is that there is one sheep in England and one half of it is black."