What does Hilbert's Satz 90 mean?

[twotaileddemon]

Hilbert's Satz 90 states that,
“for a finite cyclic field extension K/F with Galois group (S):
An element b є K has norm 1 (with respect to the extension K/F) iff b = S(c)/c for some c є K*.

Can someone please explain this to me in simpler terms? I'm not sure what he means by norm 1 or S(c)/c.

 

[jkeegan5]

can someone explain this in simpler terms

 

[mathwonk]

you don't seem to know the meanings of the basic words in the subject, so it is odd you are asking the question. it also makes it hard to answer since one does not know what toa ssume. however if you know what a galois group is, then the basic fact about the norm is that is a multiplicative function from the (non zero elements of) the big field to the (non zero elements of the) small field, which is constant on galois orbits. I.e. it has the same value on c and on s(c) where s is an element of the galois group S.

(Is this your confusion, did you mistake the group for an element of the group?)

anyway, it follows that the norm of the two elements c and s(c) must be the same, so by multiplicativity, the norm of s(c)/c must be one. The theorem says this is the only way to get norm one in the cyclic case. i.e. if the norm of an element is one, then that element can be expressed as the quotient s(c)/c of two elements of the same galois orbit.

you can read this in presumably any algebra book, e.g. lang's algebra. indeed i am only repeating what i just read there.

 

[mathwonk]

recall a galois group acts on a field extension in such a way as to leave fixed all the elements of the smaller field. conversely, in case of a galois extension, any element of the larger field extension which is left fixed by the whole group actually belongs to the smaller field. so to map an element of the larger field down to the smaller field we just need to produce an element which is fixed by the galois group. the simplest way to do this is to take the element together with all its images under elements of the group and add them together. Since adding is a symmetric function, we get a fixed element, hence an element of the smaller field. this adding operation is called the trace. if instead we multiply, we get what is called the norm. so they are pretty trivial concepts really. one gives an additive map and one gives a multiplicative map. by calling them these odd names we make it seem mysterious to the uninitiated, and enjoy ourselves at their expense, much as children hunting "snipe" do. in psychology of pathology this is called "schadenfreude", or taking pleasure in the misery of others.