Thanks! I haven't been uploading recently but will get back into it soon. If there is a specific problem or topic you would like to see please let me know
@@coconutmath4928 hey thanks for the reply! I have an algebra 2 exam next week and am struggling a little with finite fields… such as quotient fields with irreducible polynomials, cyclic subgroups of the Field of units and why for example there is no Finite field with 6 elements: F6. Any help would be massively appreciated!
@@mementomori5505 A finite field has to have order equal to p^k for a prime p. To see why this is, consider that every field is a vector space over its prime subfield. Furthermore, every prime subfield of a field is either isomorphic to Z/pZ or Q (can you show this?). If you're working over Q, then every (non-linear) irreducible polynomial has a root which is not contained in Q... so the quotient should give you Q(alpha) where alpha is the missing root. Does this help?
@@coconutmath4928 thank you!! This is a very clear explanation and helps a lot, a specific problem which I have been struggling how to approach is finding an explicit isomorphism between finite fields. For example F3/(X^3-X+1) and F3/(X^3-X-1)
@@mementomori5505Glad it was helpful! Those fields look like F3(alpha) and F3(beta), where alpha and beta are the representatives of X after taking the quotient. So whatever your isomorphism is it will be completely determined by where you decide to send alpha. Here is an example which might help (and is slightly more complicated): math.stackexchange.com/questions/108717/constructing-isomorphism-between-finite-field
Awesome video!
Hey! Great videos, would it be possible to make some more on Ring and Field Theory?
Thanks! I haven't been uploading recently but will get back into it soon. If there is a specific problem or topic you would like to see please let me know
@@coconutmath4928 hey thanks for the reply! I have an algebra 2 exam next week and am struggling a little with finite fields… such as quotient fields with irreducible polynomials, cyclic subgroups of the Field of units and why for example there is no Finite field with 6 elements: F6.
Any help would be massively appreciated!
@@mementomori5505 A finite field has to have order equal to p^k for a prime p. To see why this is, consider that every field is a vector space over its prime subfield. Furthermore, every prime subfield of a field is either isomorphic to Z/pZ or Q (can you show this?).
If you're working over Q, then every (non-linear) irreducible polynomial has a root which is not contained in Q... so the quotient should give you Q(alpha) where alpha is the missing root. Does this help?
@@coconutmath4928 thank you!! This is a very clear explanation and helps a lot,
a specific problem which I have been struggling how to approach is finding an explicit isomorphism between finite fields. For example F3/(X^3-X+1) and F3/(X^3-X-1)
@@mementomori5505Glad it was helpful! Those fields look like F3(alpha) and F3(beta), where alpha and beta are the representatives of X after taking the quotient. So whatever your isomorphism is it will be completely determined by where you decide to send alpha.
Here is an example which might help (and is slightly more complicated): math.stackexchange.com/questions/108717/constructing-isomorphism-between-finite-field