THE ALGEBRA OF SUMMATION NOTATION
... The "i=" part underneath the summation sign tells you which number to first plug into the given expression. The number on top of the summation sign tells you the last number to plug into the given expression. You always increase by one at each successive step. For example, ...
... The "i=" part underneath the summation sign tells you which number to first plug into the given expression. The number on top of the summation sign tells you the last number to plug into the given expression. You always increase by one at each successive step. For example, ...
Notes on Galois Theory
... For simple extensions, the converse to Lemma 2 is true. In fact, we can say much more. Lemma 3: Let α be an element in an overfield L of a field K. Then: K(α)/K is algebraic ⇔ α is algebraic over K ⇔ K[α] = K(α) ⇔ [K(α) : K] < ∞. Moreover, if α is algebraic over K and f (X) =Irr(α, K), then there ex ...
... For simple extensions, the converse to Lemma 2 is true. In fact, we can say much more. Lemma 3: Let α be an element in an overfield L of a field K. Then: K(α)/K is algebraic ⇔ α is algebraic over K ⇔ K[α] = K(α) ⇔ [K(α) : K] < ∞. Moreover, if α is algebraic over K and f (X) =Irr(α, K), then there ex ...
Simplify Expressions of the Form a Simplify Expressions of the Form a
... Example 5: (Simplifying expressions with negative rational exponents) Simplify each expression. Assume that x can represent any nonzero real number. ...
... Example 5: (Simplifying expressions with negative rational exponents) Simplify each expression. Assume that x can represent any nonzero real number. ...
Inside Algebra Correlated to the New Jersey Student Learning
... describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 +(2xy)2 ...
... describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 +(2xy)2 ...
De Moivre`s Theorem 10
... You should obtain 1 + i = 2(cos( + 2kπ) + i sin( + 2kπ)) k = 0, ±1, ±2, · · ·. ...
... You should obtain 1 + i = 2(cos( + 2kπ) + i sin( + 2kπ)) k = 0, ±1, ±2, · · ·. ...
Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.