Lecture 6 1 Some Properties of Finite Fields
... Claim 1 Let K and L be finite fields with K ⊆ L. Then, L can be viewed as a finite-dimensional vector space over K. Proof Idea Addition in the K-vector space is the addition law in L and scalar multiplication of an element α in L by an element c of K is defined to be the product cα as multiplied in ...
... Claim 1 Let K and L be finite fields with K ⊆ L. Then, L can be viewed as a finite-dimensional vector space over K. Proof Idea Addition in the K-vector space is the addition law in L and scalar multiplication of an element α in L by an element c of K is defined to be the product cα as multiplied in ...
SELECTED SOLUTIONS FROM THE HOMEWORK 1. Solutions 1.2
... 1.3, 13 Establish that if the set {v1 , v2 , v3 } is linearly independent, then so is {v1 + v2 , v2 + v3 , v3 + v1 }. Proof. We’ll do this using the contrapositive. Thus, assume that {v1 + v2 , v2 + v3 , v3 + v1 } is linearly dependent. By definition, this means we can find a1 , a2 , a3 ∈ R such tha ...
... 1.3, 13 Establish that if the set {v1 , v2 , v3 } is linearly independent, then so is {v1 + v2 , v2 + v3 , v3 + v1 }. Proof. We’ll do this using the contrapositive. Thus, assume that {v1 + v2 , v2 + v3 , v3 + v1 } is linearly dependent. By definition, this means we can find a1 , a2 , a3 ∈ R such tha ...
ROW REDUCTION AND ITS MANY USES
... then the equation has no solution. (2) Otherwise solutions may be found by iteratively solving for the variables from the bottom up and substituting these into the upper equations. (3) Variables whose columns have no pivot entries in the echelon matrix are free, and solutions exist with any values c ...
... then the equation has no solution. (2) Otherwise solutions may be found by iteratively solving for the variables from the bottom up and substituting these into the upper equations. (3) Variables whose columns have no pivot entries in the echelon matrix are free, and solutions exist with any values c ...
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... The idea is the following. We cross our fingers and pick a polynomial a(X) of degree less than n at random. This is some element from Fp [X]/(f (X)). If we are extremely lucky we might just get gcd(a, f ) 6= 1, and this already gives us a non-trivial factor of f and we are done. Hence, lets assume t ...
... The idea is the following. We cross our fingers and pick a polynomial a(X) of degree less than n at random. This is some element from Fp [X]/(f (X)). If we are extremely lucky we might just get gcd(a, f ) 6= 1, and this already gives us a non-trivial factor of f and we are done. Hence, lets assume t ...
Review/Outline Recall: If all bunches of d − 1 columns of a... are linearly independent, then the minimum
... Models of finite fields The only possible sizes of finite fields are prime powers (meaning powers of prime numbers). (This is non-trivial.) That is, there are no finite fields with 6, 10, 12, 14, 15, 18, or other composite non-prime-power number of elements. Fields with prime numbers p of elements ...
... Models of finite fields The only possible sizes of finite fields are prime powers (meaning powers of prime numbers). (This is non-trivial.) That is, there are no finite fields with 6, 10, 12, 14, 15, 18, or other composite non-prime-power number of elements. Fields with prime numbers p of elements ...
sections 7.2 and 7.3 of Anton-Rorres.
... sometimes attracted so many students that opera glasses were needed to see him from the back row. Schur’s life became increasingly difficult under Nazi rule, and in April of 1933 he was forced to “retire” from the university under a law that prohibited non-Aryans from holding “civil service” positio ...
... sometimes attracted so many students that opera glasses were needed to see him from the back row. Schur’s life became increasingly difficult under Nazi rule, and in April of 1933 he was forced to “retire” from the university under a law that prohibited non-Aryans from holding “civil service” positio ...
Chapter III Determinants of Square Matrices Associated with every
... Note that although there is a zero element along the diagonal of A, its determinant is not zero. In other words, property 7, above, does not apply in this case since the matrix is neither diagonal nor upper/lower triangular. Group 3: ...
... Note that although there is a zero element along the diagonal of A, its determinant is not zero. In other words, property 7, above, does not apply in this case since the matrix is neither diagonal nor upper/lower triangular. Group 3: ...