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# Abstract Algebra

From pure math to modern cryptography to organic chemistry, abstract algebra underpins much of contemporary research. Learn the theory of groups, rings, and fields.

What real number does not have a multiplicative inverse?

**Definition.** A real number \(b\) is called the *multiplicative inverse* of a if

\[a \times b = b \times a = 1.\]

What number \(a\) satisfies \[3 + a \equiv 0 \text{ (mod 7)?}\]

**Definition.** *Addition modulo 7* is defined by adding two numbers, dividing the sum by 7, and taking the remainder.

For example, \(3 + 6 \equiv 2\) (mod 7), since 3 + 6 = 9 and the remainder when 9 is divided by 7 is 2.

What number \(a\) satisfies \[3 \times a \equiv 1 \text{ (modulo 7)?}\]

**Definition.** *Multiplication modulo 7* is defined by multiplying two numbers, dividing the product by 7, and taking the remainder.

For example, \(3 \times 6 \equiv 4\) (mod 7), since \(3 \times 6 = 18, \) and the remainder when 18 is divided by 7 is 4.

For which value of \(n\) does \[5 \times a \equiv 1 \text{ (mod n)}\]

have a solution?

**Definition.** *Multiplication modulo n* is defined by multiplying two numbers, dividing the product by n, and taking the remainder.

For example, take \(n = 6.\) We have \(5 \times 5 \equiv 1\) (mod 6), since \(5 \times 5 = 25, \) and the remainder when 25 is divided by 6 is 1.

Does every real number have an additive inverse?

**Definition.** A real number \(b\) is called the *additive inverse of* \(a\) if

\[a + b = b + a = 0.\]

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